If you only need to move 360 degrees away from the starting point, I'd suggest using a fast servo with a 2:1 ratio between the servo output and the thing you want to drive, so 180 degrees on the servo gives you 360 degrees on your driven cylinder. Getting a servo with an adequate combination of torque and speed could be problematic, but I think you have a chance.
On the other hand if you need to move an arbitrary angle away from the starting point you have a much harder problem. Accelerating, decelerating and getting any sort of accuracy on the resting position is going to be quite challenging in that sort of timescale. In that case I'd try to come up with some sort of mechanical drive system that moved in increments of 60 degrees so that you just need to cycle it through the required number of steps and not try to interactively control the position and acceleration of the motor.