Apes have some of the best adapted limbs for climbing. Few animals of our size are anywhere near as good at cliff climbing as we are. We have hands that are flexible, the can grip. The can be flattened to insert into crevices, then flexed to jam them into those crevices and with moderate training we have muscles powerful enough to pull ourselves up. Our hands developed for tree climbing, but they do a good job of rock climbing too.
So, drawing my inspiration from nature, something hand-like would be a good start, and hand-like feet too. I might make a few changes. Our hands only bend one way. It would be good if the hands could bend both ways. Also our hands are easily damaged by sharp rocks, and rock climbers often need binding on their fingers. So a tougher skin would help. You would need lots more muscle power too, even more than chimps have. An extra thumb would also be useful in some situations. A couple of extra arms could also be handy (haha - pun).
For a stable hold even during sleep you would need some way to insert the hand into a crack, flex it to make it grip and then lock the position in such a way that muscle tone is not required to maintain it. Lockable positions are possible (we have lockable knee joints). Something similar could be imagined. As noted in comments, perching birds have locking flexor tendons in their feet, which is why they can sleep while perched, bats do something similar with their feet.
So the creature has long and powerful arms (six of them), and 6 finger hands (thumb-four fingers-second thumb) which can close to a fist on both sides. Abrasion resistant skin and the ability to lock their fingers in a position to allow them to hang from a crack in the rock.
They also mature and breed young and often because this is a harsh environment where death from rock falls and accidents is frequent.