I am sorry to say that you would be very lucky to find a studio in the budget and location you are asking for. March 1st is a little premature to be looking for, but use the time to look for anything now, and if you find it take it, even if it means paying a couple of months rent that you don't want to pay.
This is traditionally the slowest time of the year for rentals, with the least number of people looking. The Spring market turns very fast and keeps heating up with the weather through the Summer.
But getting out there and educating yourself (looking) is the best way to find a space that works for you.
It will probably mean moving to an hour or more commute for the budget.
Suggested neighborhoods: Eastern Bed Sty, Crown Heights. Ditmas Park, Borough Park, Sunset Park.
All the best,
Kathryn Lilly, Broker