I live in NYS with my girlfriend who asked me to come stay with her from out of state. We are in a rental apartment complex but I am not on the lease and the rent is paid through the end of the month and she wants to turn her keys in and leave. I have mail going there, the cable is in my name what rights do I have the right to stay till the end of the month?

First, she can't just leave. The lease and the law requires she give notice, of at least 30 days. So even though the rent may be paid until the end of March, she will owe for April at least. This assumes that she is on a month to month lease. If the lease expires at a later date, then she may be responsible for a longer period.

On the other hand, since you have established residency, the complex may be willing to have you sign a lease. I would talk to them