Welcome to New York!

I recommend looking on StreetEasy and Zillow for rooms for rent. Focus on the less expensive parts of the city (i.e., anywhere outside of Manhattan below 96th Street, Morningside Heights, western Queens and northern Brooklyn). Widen your net. It wouldn't be the end of the world to land in Jersey City or even St. George. Here for instance is a $1,400/month studio in St. George that's available right away.