The application deadline for the public pre-K will have passed before you get here (I think they are around Easter).
There aren't that many public pre-K spaces and most are taken by siblings. You would be very unlikely to get a public pre-K spot in your zoned school even if you were able to apply in time. NYC does not have to to provide pre-K places.
The private or church pre-K's usually have application deadlines as well. Many have open houses from Oct-Dec and you have to apply by then, or you apply in January and put your deposit down in February.
However, that being said, often there is a lot of movement as it is a big city and you should be able to find a pre-K program of some sort you like, it may not be your first choice. You may have to take afternoons rather than mornings, etc.
Many schools are closed for August so you would not be able to get much information then.
The challenge for you is finding somewhere to live and then finding a pre-K program nearby.
Also when is your child's birthday, the school age runs Jan - Jan unlike where I was from in the UK Sep-Sep.
Sorry, it's turned into an essay! There is a woman who blogs about it who has quite a bit of info for Brooklyn called nycschoolhelp.com, you could email her or call her as well, I think she charges but info on her website.