This is decent, but needs some work. The main thing I'd think it needs is some proofing for typos and evening out the language. A lot of sentences feel weird or drag on too long. Stuff like
neaking past the city guard was the easy part; it was actually leaving town that was difficult: huge chains suspending midair between the town and a bottomless pit aren't the simplest to traverse.
Would flow a lot better if cut up into multiple sentences.
He gripped the artery in his arm, slowing the bleeding down. He thought quickly and tied a piece of ripped undershirt around the wound; another piece securing itself around the pulsating vein.
Veins don't pulsate. Also, I imagine this would be really hard to do on your own arm (by which I mean I've tried it, and it's nearly impossible to get it tight enough to matter). A hand injury probably wouldn't bleed enough to need a tourniquet and risk killing the limb anyway.
I'd like more vivid descriptions of the areas he's in. There's nothing in here that particularly evokes interesting imagery or paints a good picture of his surroundings.
Gonna pull a vivax here and say that I'd like more of a sense of who this character is and what his motivations are.
Stuff like
The ground wavered. The earth faltered. The floor fell. Swallowed by the city, he fell; not a smooth fall, not in the least. Smacking against the jagged concrete faults every now and then, he felt intense pain with every blow. A final blow to his mask switched everything to black.
deserves a lot better description than it gets. How did the ground rumbling feel? What did it sound like? How did it look?
In general this could stand to be lengethened. You move through events too quickly so the reader really never gets any emotional reaction or feel for them.
Not a fan of the ending, unless this is going to be getting an ending. Leaving things open is good, but this doesn't really pose interesting enough questions to make me keep wondering about them after the fact.
Overall you've got a good start here, it just needs work to push it to a higher level.