First, get off the idea this will happen in a month. No coop is closed in a month. I bought on the private market and it took 8 months.

Second, contact UHAB. We sell sponsored apartments all over the city for $35,000 and up. Wed do this for people who can put down an initial payment of around $5,000, and have decent credit, proof of income, and interest in the purpose of HDFCs, that is, a limited equity coop that they will not expect to cash in on at market rate some day.

The coops we sell are limited equity, so for each year you live there, you get 3% more than the value you bought it at. That means it creeps up a modest amount, but you do get some return. Also, it keeps the price down for the next person in line when you move or upgrade, so they can also benefit from this truly precious gift of permanently affordable housing.

You can also see if you can find a private listing for coops out our site, but you are an ideal candidate for our sponsored units, which are all newly renovated (though there's less buyer choice).

Aileen is right. There aren't many HDFC apartments in Harlem in your range. There are two for sale on 530 Manhattan Ave for $130 K each. On one is an accepted offer with contracts out, the other is still available. Sunday's open house is from 12:00 to 1:30 PM.
Apartment 57. If you need more info on that one, please email you and I send you the information, I am the listing broker.
First time buyer is okay, $40 K is okay, downpayment is perfect (90 % financing on this one), just do the pre qualification.

You should get pre-approved for a mortgage before you start making any offers.