Baboons have the best vision of any animal (mammal) on earth, or so we were taught in School here in RSA.
After working as a PH now so many years I have to agree that nothing I've ever tried to stalk has the vision capacity of a Baboon. If you can stalk within 300 yards of a troop your doing really well! They will usually post a guard in a tree to watch for movement while the others are foraging. That baboon is always a Male and usually the number 2 or 3 dominant one. He's the one to shoot if you can get him. The biggest dominant male may also be in a tree but usually with the females "poking around" or sitting nearby keeping track of his harem.
Shooting them near water is the most common area to hunt, or shooting at long range. Baboons have AIDs. It's belived to the the origin of the problem. Getting bitten by one is a death sentence, almost guaranteed. You do not want to be tracking these things when wounded! They move like lightening and have 4 thumbs to grab and climb with all 4 feet. Their canine teeth are about 50% longer then a lions teeth too.
Any .22 centerfire would work fair. These guys weigh about 50 pounds for an adult male and I know I've seem huge males that would be a fair bit bigger then that. On average a troop would consist of lots of 35-45 lb specimens. Mostly feamle and young with a few teenage males mixed in too.
My choice would be a 25/06 or 6.5mm with Nosler Ballistic tips. That would be flat enough to reach out with an explosive bullet to cause massive fatal trauma and eliminate looking for the wounded ones. A blown up baboon is better then no baboon at all!