Lease Takeover - Okay to raise the rent?

Not sure if lease assumptions are how it would work in a stabilized situation. More likely the landlord would want to end the existing lease and then have you sign a new one, called a 'vacancy lease,' because a vacancy means they get to bump up the legal rent by a big chunk (over and above the small annual increases).

They can choose to charge you less than the legal rent (your actual rent is called a preferential rent), but under current law the preferential rent is only in force until the next renewal, at which time the rent can be adjusted to any other number below the latest legal rent.

Read in depth at http://www.nycrgb.org.

What neighborhood?

when you start a new lease the rent goes up by a vacancy increase. The current tenant could sublet to you, which they are legally allowed to do and the owner can charge you a sublet increase percentage. All of this is detailed in the rent stabilized guidelines detailed in the link provided by uptown joe