Congrats on getting such a great number! In general, this is not a short process, and it has many twists and turns, and you will be kept in the dark for much/all of it.
A number of factors go into when and if you will be called to submit documents. In fact, that is my first question: Have you been asked to submit documents?
Even if you have been, I would say the odds are not good that you will be processed in time for July 31. Here are the factors that will affect your situation:
Does this lottery have a community preference? If so, are you currently living in the same community board as the building? If the answer to both these questions is yes, you will experience priority processing—yay! This is a great situation to be in.
Please see below for a caveat!
If this lottery
has a community preference but you are
not in the community board, then all the people in the CB will be processed ahead of you for 50% of the units. That means that although you will be called at some point, because your log number is so low, it's hard to predict when.
If this lottery does
not have a community preference, you are in good shape with your low log number.
Another unknown factor is the completion of the building, assuming it's under construction. Even if you are processed successfully, you cannot move in until the building has a certificate of occupancy. Sometimes this takes months. To find out some hints, you can look up the address on the department of buildings website and attempt to suss out which tasks remain to be done. (It's not really easy unless you're in the construction business.)
If you are lucky, the building will have a C of O already.
Here's where you can plug in the address:
https://a810-bisweb.nyc.gov/bisweb/bispi00.jsp
Caveat: If you do have community preference, you must be within the CB both for when you're selected and when you're processed. So if you need to move in temporarily with family or friends, you should try to stay within the CB so that you don't have to lie about your address. Any lying will get you disqualified, or worse.
Best of luck!!