As seems so often with this kind of thing, the answer is hidden in the question. The frictionless system, weightless rope etc. Everything is designed to have them both reach the top at the same time. Real world, I think it would depend on how the monkey climbs the rope.
Very slow climb with minimum displacement of centre of mass and the friction in the system would allow the monkey to get to the top with minimal movement of the weight.
Significant displacement of centre of mass, use of the monkey’s tail to provide rotational acceleration in forceful yanks on the rope and I think the monkey could get the weight to the top without progressing very far in either direction.
Of course, it’s a perfectly frictionless system and the rope has no weight (or mass!?) and the monkey probably doesn’t swing around or behave like a monkey so…
The solution will probably contain the phrase “equal and opposed reaction”.
(I’m not sure if a/diabatic evolution is the correct term to use here, but that’s what struck me off the bat for the real world scenario)
