What you're expecting your character to do would be called a "muscle up" at the gym, i.e. a pull-up followed by raising your entire torso above the bar. Only highly-trained, elite athletes can do these, e.g gymnasts. However, they are not impossible and go on you tube and you'll be able to find videos of men and women doing these kinds of exercises, including with weights on. One-armed though... I don't know how many people can do this... probably not many but It's possible that someone, somewhere can. It would take a hell of a lot of training though. You tube might have videos of it, if anyone can do that. Weight is a factor. You need a lot of upper body strength relative to your body weight to do these kinds of exercises.
For a highly trained soldier or athlete, male or female, I would not find it implausible that they could muscle-up (2 handed) from some kind of sturdy ledge to save themselves from falling. Cliffs are a bit more precarious because the rocks are crumbly. This may vary by geographical location, but in Britain, cliffs tend to be made of rock that breaks easily and I've had warnings drummed into my head since childhood about not walking too close to the cliff edge in case the rock I'm walking on breaks off. People can die just from standing on a cliff edge.
One-armed muscle ups stretch plausibility a lot though, especially if the character's dressed in full armour at the time. In a futuristic fantasy setting you could mitigate the weight factor by having the armour made of some ultra-light, very strong composite material or something, but for historical fantasy settings, it's going to be heavy. If you go this route you'll have to previously establish that your character is much stronger than even the other highly trained soliders, and then you're in possible Larry-Stu territory. You'd have to write such a character carefully to not end up making him a Larry-Stu.
Having said all that, it would be a lot easier and more plausible for the character to use his legs to get himself up. Legs are way stronger than arms. If he can get even a small foothold on the cliff face and push himself up with a leg, he doesn't need to have super-elite athlete levels of strength. The average reasonably fit person will have strong enough legs and it won't take any more leg strength than going rock climbing.
IMO if you want to have him rescuing himself from dangling from one hand from a cliff (assuming it's a reasonably vertical cliff), first he can observe that it's lucky that the piece of rock he's holding on to looks sturdy - he can't tell for sure and hopes hard that he's right - then he looks for a foot hold and manages to swing his foot into position. This means his weight is distributed between the rock he's gripping on and the foothold (less likely that the rock he's holding on to will break off). Maybe the boots he's wearing have good soles for gripping, which helps him keep the foothold. Using leg strength he pushes himself part way up, giving him leveridge to get his other arm onto the rock he's holding on to. Maybe he finds another foothold so can push with his other leg too. Now he's most of the way up and is able to use both hands to pull himself the rest of the way up. I'd find something along those lines plausible, even in heavy armour. I would assume that your character is stronger and fitter than average just by the fact he's a soldier/warrior.