Here is a solution for a simpler case like this one:
- Create 3 objects: The cylinder, the rope (a cylinder with an Array modifier and a Curve modifier) and the curve.
- Rotate the cylinder with some keyframes (here -400°). Move up the curve, move down the rope (or increase its Array count) so that the rope seems to follow the the cylinder. Make sure that each of your animations interpolation is linear (in the Dope Sheet press T > Linear).
You can find more at Animating wire/cable/rope wrapping around a cylinder