According to
http://sls.downstate.edu/admissions/est_residency.html:
"The deadline for change of residency by matriculated students with appropriate documentation is one month prior to the registration date for your college for the next semester (you must file a change for residency status six weeks prior to the registration deadline that you wish to have the change in effect). The Application for continuing/returning students is available on-line and also in the Office of the Registrar."
Awww that's nice of them, they put up the specific date since the last time I checked it
http://sls.downstate.edu/registrar/com/calendar/2012-2013.html:
"Fri., July 26 Last Day to Apply for New York State Residency for Fall 2013 Semester" (That's Friday, July 26, 2013).
So I guess July 26th is the due date of all the forms. I guess I would say a lease starting by July 20th of this year *might* be adequate... However, there is some paperwork that needs to go into it and pretty much everything needs to be notarized (dunno if this is the type of thing that can be done in 5 seconds or not, but I figure it's best to leave a few days' wiggle room, especially if some of the documents you are using as proof are in someone else's name and you need to rely on getting copies of the bills from them or statements written by them to verify).
I am heading over at the end of June, so I guess I will be there with plenty of time!
I called the financial office and they said that 10 months lease would be sufficient for the in-state tuition during the second year if you submit the application with the letter stating why you don't have a full 12 months lease. She also told me that it is okay to move over in August and you can still get an in-state tuition after first year.