The previous post was pretty well on the mark. Let's just take a look at some specific areas. Upper body pull, definately. Chins, rows, etc. Some upper body push to balance it out. Also, don't forget the posterior chain and quads, it will really help for pushing off with your feet to assist your upperbody when pulling yourself up. Any type of squatting, lunges, step ups, Deadlifts etc, and as was previously pointed out, your core. As Matt stated, you definately don't want to get bulky. and bodyweight stuff is good, so are KB's and DB's. Strength/ndurance would be key here as well. You might want to check out http://www.crossfit.com
It's a general fitness site, lot's of sevicem,en, fireman, martial artists, police types over there, with a large variety of interests, and have seen some rock climbers.
They focus on about all aspects of fitness, and are very heavy into strength endurance.
Good training. You shouldn't expect to get through these workouts as written, just scale them back. I do a lot of this stuff and am 56, so at 52, I'm sure you can benefit from it.