Rate of star collpse It is believed that our sun will become a red giant with a diameter of about 1 AU. When the fusion slows down, gravity will collapse the sun. Since the energy release from the fusion diminishes slowly (maybe hundreds of years for a significant change of 1%). 

Let's assume that only a small fraction of the sun (10%) is needed to collapse (to 1/1000 the volume) to produce a nova. Would it still be possible for a star to collapse in a few hour (instead of decades)?
Well, I'm not sure exactly what you mean by "collapse" (if it takes $10^9$ years to die, the last few hours satisfy what your asking about). However, maybe some explanation would help:

You're asking about the interaction between two timescales - the Dynamical Timescale, which is the time it takes for a star to free fall into a nova, and the Nuclear time scale (I think), which is the rate at which the sun exhausts it's fuel. The dynamical timescale for the Sun is like 1000 s, and the nuclear timescale is like a Gyr (usually what we think of when we think "lifetime of the Sun").

Any nuclear effects would totally dominate over dynamical effects (since they take much, much longer), and without some specific collusion of behavior (angular momentum, metalicity, what have you), collapse would occur with the burning of fuel and not with the dynamical timescale until the Sun exhausted all it's fuel, and would collapse in a matter of minutes.