Check between 86th and 96th between First and Second.
Lots of 5 story walkups.
I think with enough shoe leather you can find an $1800 apartment.
Not to recommend anyone but here are a couple listings in the neighborhood. Maybe old listings but it gives some idea.
LISTINGS
But walk the streets and take numbers of management companies, ring supers bells, etc. Last week of the month talk to anyone moving out and ask what apartment they are leaving and what they paid.