The mechanics of it is all going to be pure speculation, since it's not explained in the film.
At minimum, we know his memories are being replaced. Muscle memory is also in the brain (just another type of memory), so it makes sense, given we know his memories are being updated, that muscle memory also comes along with that.
We can also safely assume his blood is also changed in some way, since bleeding out would give the Omega back its reset ability. The alien blood infects him in the future and somehow also physically infects his body from yesterday. Potential side-effects of having the alien blood in his system, like, say not needing sleep, are unknown, but possible if the writers decide to expand on it at some point.
Otherwise, I think we're supposed to assume he wakes up physically the same each and every time, just with new memories. This heals wounds and puts him back in a refreshed state. It's the morning, and he's already had a good night's sleep (and a taser nap).
And while I'm sure he and Rita might pull all-nighters on occasion, he doesn't always have to do the same thing every time. He has plenty of opportunity to catch up on sleep if he wants to.