You need to consider a room with such budget. To rent a studio for less then $1000 in NYC is almost impossible.
Coming back to most affordable neighborhoods, I advise to look for rentals in Sunnyside and Bay Ridge. the first one is about 15 minutes from midtown Manhattan. And it can take almost 45 minutes via the train to get from second neighborhood to Manhattan.
If you find an apartment that you like, but the rent is hight for you. Try to negotiate the price with your landlord. Fortunately, there is a rental application platform https://rentberry.com/custom-offers that offer to submit the price for vacancies online.