There are a LOT of things you need to figure out first!
1) Are dogs allowed in your apartment
2) Are Pit BUll type dogs allowed in your apartment
3) What are the extra fees for the apartment with a dog
4) Are you willing to spend an hour a day physically exercising your dog?
5) Are you willing to spend an hour a day mentally exercising your dog?
6) Are you willing to spend 30 min-1 hour a day training your dog?
7) Are you aware of the vet/medical expenses with a puppy? What about anually?
Are you able to walk the puppy multiple times a day? (for however many months old the puppy is, add an hour and that is how long they can hold their pee/poop.
As for your initial question on a pit bull living in a small apartment...yes it can be done and many of us here have done at some time or another. The key is how you answer ALL of the questions above that will determine if you are ready for a pit bull....much less in an apartment.