Michael Schoch, Answers questions on Noodle
It looks like the application periods for pre-k in New York City were on April 24th and July 10th. You can try calling P.S. 161 Arthur Ashe School at (718) 441-5493 to see if there is space available in any of its classrooms.
You can look for other pre-k programs in NYC on the Noodle Preschool Locator.
If you're interested you can read more about how preschool works in New York on this handy, state-by-state guide.
Best of luck! Feel free to ask a follow up if you need!