Hey there Fox
Here is your requested review.

I do hope that what I say now helps. I'm in a particularly good mood, so I'm going to take it piece by piece before I look at it overall.
I knew how my life started.
As a first sentence, while this is not remarkably week, it is not wholly gripping either. The first sentence should start the spin of the story, induce us to read more, and most importantly set a tone for your story that defines who you are as a writer. You haven't told us much here, and I do believe that we would be missing nothing if you merely neglected to tell us this sentence at all. You want to make a difference with every word that you write. If it doesn't help build our picture, don't write it. Okay, that's the end of my rant on first sentences.
It began with a moment of joy, a feeling of no restraints and disturbances.
I'm not sure how aware of this you actually are, but what you've said here is that you had a feeling of no "restraints and disturbances", which would imply that you felt no restraints
and disturbances at the same time. You might have however felt restraints without disturbances or vica-versa. While this is not a critical error, I thought I'd use it to illustrate a point: you want to be as clear as possible in every circumstance. In this case, substitute the "and" for an "or".
Though there were
struggles, there was still joy.
To a reader this seems quite paradoxical. What I mean by this is that you told us that you felt no disturbances, but now you tell us that there were struggles, and that the joy was despite that. My feeling is that if you show us what actually happened immediately, you don't need to explain what the emotions were. My advice to you, is to show us what actually happened, and let us see the feelings ourselves.
Like a baby being supported by her mother after she fell to the ground
trying to walk, I felt joy.
By now you've overdone the "joy". You've described it as well as it needs to be described before this point.
It was an art itself, a number of pictures that moved and cannot be described.
This metaphor is remarkably tough to swallow. Even the Mona Lisa wasn't an art. It was an artwork. Once again, be very specific. But better still, in this situation you can drop the whole thing. Replace it with one sentence like, "In the beginning, I was consumed by a feeling of an indescribable joy." This is brief, sharp, and tells us what we need to know.
However, as I grew up, the joy started to dwindle.
Switch the word joy for another.
Slowly and painfully, it was torture in itself.
It "became" torture itself.
Now, watch closely what I do with the next piece:
I knew how my life would end. It would not be like a sword through the neck but a sword through the side, slow and painful.
Several problems exist in this little piece. Firstly, to start off with we don't know what you mean by a sword in the neck. We are only exposed to its implications when you contrast it with a sword in the side. We need to understand your comparisons to start off with. Secondly, we don't feel the penetrating emotion that you were aiming for. Thirdly, the second sentence here is wordy and tough to read. For me, this all filters down to one problem. You need to expose your paragraph appropriately. This means telling us things at the opportune time to maximize our emotion and allow it to leave an imprint on us building up to a climax. I'll give you an example:
I could already feel the life being pulled away from me. Perhaps a quick painless death would have been welcome. I knew, though, that mine would be slow and painful. It was like feeling the blood oozing out of a stab wound in the ribs, anticipating death. I knew how my life would end.
It started on the upper floor of the okiya where displays of kimonos decorated the room.
Be careful, you're telling us that "it started" too many times. Rather tell us the story itself.
The floor was golden and sliding doors stood at each side of the living room, one to the veranda and one
that lead downstairs.
I'm going to criticize you here for dumping information in large chunks. Always try to weave your description into the story itself. It allows us to take in some of the details rather than just read through them.
I sat down before a small table opposite my guardian wearing an elaborate blue and grey kimono
that flowed freely from my shoulders. My hair was decorated and white makeup covered my face. I was
like the snow at winter on the gardens of Kimi-ku.
Often we don't really want to know what you are wearing. I would refrain from telling us unless there was a specific reason that you noticed it. Remember, when writing in the first person become the person. Describe what they would and no more. No less as well. Remember also that your job as narrator is to tell us a story. Not to show us something we wouldn't want to know about.
Okaa-san told me that when drinking a porcelain tea cup, a woman must raise her arms slightly
high so that the man would see my underarm.
I'm not too well versed in Japanese custom, but as I understand it, the term San is far less formal than the term "Sama". I do believe the latter would be a more appropriate form of address in this case.
She held the tea cup firmly before me after but had not raised it high for she told me that this is a mockery to other geishas.
was a a mockery to other geishas. Also, you might want to include some dialogue here. Dialogue is one sure way to introduce us to who your characters are, and I would like to see a little more of that.
I gave a firm bow before I raised the tea cup placing one hand gently at the bottom while my index
finger held the handle.
The problem with this and in fact quite a lot of your story, is that you tell a lot of what happens, but you don't show us how it happens. Be more descriptive in your narration itself such as, "I bowed my head before I curled my index finger through its handle and slid my other hand beneath it. I raised the teacup in the air, replicating Okaa-San's actions.
Slowly I began drinking it but at the middle of my procedure,
I'm not sure what you mean here. The word procedure doesn't fit quite so well here. Describe what happens in simpler words and place more focus on their placing rather than the words themselves. Also I do believe that the sentence should end here.
the cup broke so suddenly. Its pieces scattered on the floor.
There should be a comma rather than a period here.
I gave a number of bows for an apology even scratching my skin against the pieces of glass.
You didn't give your bows in apology. You felt remarkably bad, so you felt yourself bowing over and over again in rapid succession. Try to show us what your response was.
Okaa-san commanded me to sit up and gave a sharp gaze at the pieces.
You can't "give" a gaze "at" anything. You need to gaze at something. This should be, "Okaa-San commanded me to sit up, gazing sharply at the shards of glass scattered across the floor."
I was about to bow again but she ordered not to.
Do not tell us you were about to. Show us what she says. Eg. "Stop bowing!" she said, "it wasn't your fault... etc."
“It was not your fault. It was just bad luck.” She told me but I felt that it was my fault.
When you tell us what you feel, you need to defend that feeling. Why did you feel that it was your fault?
The blood that flowed from the tip of my finger approved of that.
approved is the wrong word here. It "proved" that.
I never really understood what she had said but it was at that exact moment that my life would change. It was at that moment that I learned about the Tournament.
You've defined an exact moment, but you haven't made it that exact. When exactly was this exact moment? When you broke the cup? When she spoke to you? Two minutes after that?
Rain poured over me as I walked hurriedly through the dense streets of Gion a few weeks after
the incident.
Why were you hurrying. We need to know. If you don't like water, tell us that.
Several people walked silently through the rain.
They had not spoken to each other.
If they were silent, then they certainly hadn't spoken to each other, right?
They simply walked alone.
But they're not alone if they are in a group, right?
“Excuse me, Mister, you’re hat…”
*your. Remember, you use the apostrophe to contract two words, in this case "you" and "are".
I called out his name
His name is "Mister"?
after I had bumped his straw hat into the
streets.
You want a little bit of chronology. Tell us she bumped his straw hat before you show us the dialogue.
His small bent back was still visible from the crowds that encircle him.
Try to keep to one tense. "encircled".
I tried chased him but it was pointless.
You need to explain your motives. Why did you try to chase him?
I gave a sigh before I continued walking to the okiya.
This is for me probably the most common mistake I see when reviewing first person narratives. You never account what you wouldn't notice. Sighing is a subconscious action. Don't tell us that you sigh unless there is a tremendously good reason for you to notice it.
The first thing I heard when I went inside my okiya, was a scream.
Think how much more effective this could be if you said this in the active. If you don't know how to do this, there's an article on "Writing Gooder" about it. Here's the link:
http://www.writinggooder.com/2009/06/24 ... ive-voice/ I stood by my wooden
doorway dumbstruck, as the darkness swallowed my body whole.
Do try to explain your motions. Shorten your sentences and suspense will build far more easily.
It was an unwanted shock. It was an intrusion, a stab to my heart. I held my hand
close to my beating heart. I called her name.
Firstly, we know it was an intrusion, and why was it a stab to your heart? Usually that would only be felt after you knew she was dead. Also, you repeated the word "heart" unnecessarily.
“Okaa-san, is everything alright?” I asked with trembling words to okaa-san, a word used for the
owner of the okiya who was my guardian.
One of the key tricks to writing dialogue is to write the words to be so emotive that you don't need to define them as "trembling" or otherwise. Also, as soon as you break away to tell us who Okaa-San was, you lose a lot of your suspense.
I took a few trembling steps into the darkness, and lay frozen to what I saw. She laid there on
the floor with her bloodshot eyes staring at me.
She lay there. Not laid there.
She called my name but I only stood still. My breathing became rapid. I felt that the whole house closed in on me to the point that I could not move. I had not felt my legs and my body was trembling with shock.
Try shorten the sentences and avoid repetition. Suspense builds better if you do this.
Each call from her, I stepped back horrified to what I saw before me.
This seems to be fragmented. Complete the sentence.
“This cannot be real!”
Remove the inverted commas and change "cannot" to "could not" and this would be marvelous! I really like your attempt at building up to emotion here.
It was only after she called my name once more, that I rushed towards her calling out her name
repeatedly hoping that she was still alive.
Surely if she called your name you knew she was still alive?
It was complete irony. I have stood still while she was still alive.
Once again, watch your tense.
Yet I had only moved when she
was dead.
Incorporate this fragment into the previous segment for better effect.
It was just before I could reach her that I was pinned down to the ground. I stared across the floor as
two people, Collectors, bounded my wrist.
Firstly, how did you know they were "Collectors"? Secondly, we want some emotion here. Screaming, kicking, and building of a real scene. Let us know you struggled. We want some more suspense.
Tears flowed down my soft cheeks. It was cold.
What was cold? The weather or the tears?
It was different from the tears that flowed before.
"they" were different etc.
I exclaimed as they bounded my mouth with a rag.
Bound is already past tense for bind. They can't have "bounded" you, they bound you.
The last thing I had seen before they covered my face in a black cloth was her blank eyes that stared
at my soul. It haunted me. It tortured my dreams into nightmares.
This is superb! I love the description and you've built a vivid image. I would make a suggestion though. Try change "that stared at my soul" to "that seemed to stare through my soul".
It made my life filled with no purpose.
This sentence doesn't quite make sense. If I were you, I'd simply chop it off, and end your chapter with your marvelous sentence above.
Okay, That's the piece by piece. Here's my more general advice:

On Prose: I would start trying to use simpler words, and form more coherent and well structured sentences. Once you have that, the more complex words will come automatically. You have quite a few fragments here that you might want to turn into full sentences. Also, try to build things up to excitement as best you can.

Description: Some of your description is great, but some of it is lacking. As a general rule: If we wouldn't understand something describe it as you go along. If we wouldn't want to know something, don't describe it. When you do describe something, try to describe it simply. Just enough to give us a picture of it. Also, use description to show us what the persona sees and transport us into their mind. If you can do this, and get us to care about the characters, you are victorious.

Passive Vs. Active: See what you can do to write things in the active voice wherever you can. It helps take us into the action.

Become the character: Describe what your character sees exactly as you think they would see it. You want to specifically focus on responding as they would to a tough situation, and communicating this effectively to your audience.

Setting: We don't actually have much of your setting. Describe what's around you. Not in one shot though because that doesn't help anyone. Describe the occasional things that your character notices, and that should build up enough of a picture of where your story takes place.

Research: I think that you want to try to research the Japanese culture as best you can. When you do this, your story looks far more authentic and legitimate.

Overall: I think you have something to work with here. You need to try to grab our attention a little more at the beginning of the chapter and work on some of the things that I've mentioned. The best way to improve at writing is to write more. The second best way is to read more. Those last two tips are more important than all the rest that I've given you here.
I hope that the review helped.
Have a great one!
"You've got to march up to this woman right? You look her in the eye, you lean forward just a little, just almost all the way, you let her lean forward just another little bit til you're just a lip distance away from each other and then you tell her - how much you hate her." - King Julian