The first thing to do is to draw a sketch.
The plane z=2 is easy to draw.
The hyperboloid is x²+y²=z²-1. Now x²+y² being positive, z is higher than 1 (or lower than -1 which we do not consider here). For any value k of z (higher than 1) the intersection of the plane z=k with the hyperboloid is the circle x²+y²=k²-1. And in polar coordinates x²+y²=r².
The volume is therefore . I let you find the boundaries for each variable.