Suddenly I remember.
I'd put a comma after "Suddenly".
I got no time to remember what it is
"I" —> "I've"
I dress up. I walk out of my house, lock the door behind me.
I'd incorporate "I dress up." into the other sentence, make it one statement on leaving. I.e. "I dress up, walk out of my house, lock the door behind me."
small yellow, cast-iron staircase.
Reads weird as is. Throw in a comma after "small".
I keep running. {1}
Your enthusiasm to cite your images is appreciated, especially by copyright moguls like 
UncertaintyCrossing I'm sure, but you really don't need to cite them here, it's disruptive of the narrative. Just cite them in an author comment. (This goes for all of the times you do this.)
on a dock
"on" —> "onto"
This sends me slamming right onto a parked taxi
Again, wording's a bit wonky. I'd suggest something along the lines of "This sends me right into a parked taxi" as there is already an implied momentum, or maybe something like "I slam right into a parked taxi". Up to you.
something that awakens the driver.
Eh, personally not huge on "something that", I'd suggest replacing it with "which" or something of a similar ilk.
I hear myself saying, as I open the left back door
Nix the comma, "left back" —> "back-left".
playing beach volley, while reggae hits
Nix the comma.
I turn my head back, to keep staring at them.
Nix the comma.
I jump out of the taxi soon as the driver brakes.
Toss in an "as" before "soon".
I give him the money from the open window and rush across the road, with the bag heavy on my right hand, to the building.
Two things:
1. Nix the "with"
2. What bag? I understand that this is supposed to be a dream and logic is a bit out the window, but even the protag makes no note of suddenly having a bag, it's just there and they're holding it. When the world changes in a strange way, the protag makes note of it, yet this bag has come in with zero mention.
behind of which stood a woman with her back turned to me.
Nix "of".
She should be around 30.
A much more personal preference, but I do, in fact, head a crusade against numbers with less than three digits not being written out. Personally, I think almost all numbers should be written out, but I understand that not everyone is as radical as I am.
"Excuse me?"
This isn't a question, it's a statement meant to bring attention to the speaker, so exchange that question mark for a period.
The label on her shirt sais
"sais" —> "says"
Which is my destination, anyway?
I'm something like 75% certain that you should nix the comma.
the central are of the wing
"are" —> "area"
Suddenly, a strange motive appears.
What you go on to describe is not a motive.
I am following thus a diagonal course.
Move the "thus" to the start of the sentence and give it a comma. ("Thus, I am…")
I bet this is a dream, the time moves irregularly.
Nix the "the".
on an greek island
"an" —> "a", "greek" —> "Greek"
I feel the urge to parkour my way down, to a turnpike I can see from here.
I'd nix the comma here too.
And I am doing it. I descend from roof to roof, front-flipping and twisting in the air, with upbeat music filling my mind. A villager or two even start clapping, as if following the music's beat.
Not huge on this part, feels real showoff-y, unnecessary.
It's a long descend
"descend" —> "descent"
It's a long descend, taking five minutes or so.
Until I step foot on the ground, near the highway.
These should be one statement, connected by a comma after "so".
I walk onto the road, and lie down on the heated asphalt.
Nix the comma.
When I wake up, I see this:
Yes, you have an image of a bus, but still describe it, say it's a bus. There's more to perception than just sight.
I try to find answers to all this questions.
"this" —> "these"
I gave the bag to her,and what the bag had.
Just a note to put a space in after "her,".
Why nobody could see me?
Move "could" to be after "Why".
General thoughts:
I'm inclined to agree with 
kblacke's crit of this, generally speaking. Yes, I understand that this is somewhat a recreation of a dream you had, but it suffers immensely because of that. In real life, a vast majority of dreams make no sense and translating it to page isn't gonna yield much storytelling. I don't find myself intrigued by the disjointed nature of the piece, I'm confused and apathetic. You can have dream-like pieces, I have nothing against those (some of my favorite pieces on this site I would label as "dream-like"), but you can't wholly run on dream logic for plot progression (or even plot, generally). Of course, all rules are made to be broken, but this isn't one of those times.
In terms of a part two, as you mentioned before: you can't wait for a part two to give this one at least some semblance of purpose or sense. There needs to be something worth hanging on to now, not later.
