estranged friend is going to abandon cat on my doorstep June

Read The "Pet Law"
http://www.tenant.net/Rights/pets.html

This is what it says:
Under the "Pet Law", if a landlord fails, within three months of his knowledge of a tenant's open and notorious harboring of a pet, to enforce any applicable "no pet" provision, then any such provision is deemed void. The law applies to unit owners, as well as renters, in co-ops and condos, whether private or government subsidized.

Has your landlord strictly forbidden pets in your apartment or does it just say no pets in your lease? Obviously what I wrote above does not apply to the former. But maybe you could keep the cat!

I can see why she is a "former" friend! Why is she abandoning the cat?

Be aware that, in addition to being cruel, it's a crime to abandon an animal. (To report animal abuse, contact the ASPCA NYC Humane Law Enforcement Department at [email protected] or (212) 876-7700, ext. 4450.)

Here is a list of local shelters and rescue organizations you can try, although they tend to be overloaded, so it's best if you can re-home the animal on your own.
http://www.animalalliancenyc.org/aboutus/participants/atoz.htm

Post a photo if you can--cute pics always help!

So this person can take the cat back after a month?? If you put the cat in a shelter there's a good chance it will be euthanized. I really don't recommend that. Would the "friend" be willing to take the cat back if you find a home for him/her for a month?

Also, even though you don't have a lease (which I just re-read), tenants have a lot of rights. So your landlord can't very easily kick you out for having a cat. If we assume you're a good tenant, then he really has nothing on you to kick you out--even a cat. I'm sure he's more worried about dogs barking and cats scratching up the apartment. Do you recall if the cat is pretty vocal or is it quiet? If it's quiet, if I were you I would take him in--at least for the month if your "friend" is willing to take it back.

Definitely post some pictures! That always tugs at the heartstrings.

Do you know if the cat gets along with other cats?

Is the friend gonna take the cat back upon her return?