john is referring to medial epicondylitis or elbow tendinitis.
Pull ups create an extreme amount of varus force on the outside of the elbow. Rock rings or dowel rods allow your elbow to pivot and not place so much stress.
John's idea is most excellent. The dowel rods or tool shafts require lots of grip strength, and the pull ups get your shoulders and arms. Training "sport specific" will increase your strength for that sport. Doing a bunch of pull ups will help those of you who are generally weak and who don't have to strength. Slow pull ups and isometric holds will train your muscles at those specific arm locations. Way more time is spent in a isometric hold climbing, than it is in the actual process of a pull up.
try not to work on grip strength specifically too much, injury is too easy in that area. By combing it with an activity like a pull up, you are doing more functional training and are less likely to injur.
To round out the training, be sure to work your wrist extensors and triceps, and also your lower traps.
A hammer swing with tubing or weights is a good way to exercise the extensors and provide a functional way to train, sport specific. Lower traps do ground angles on the carpet or wide grip lat pull downs and only pull down a few inches. Finally a punch-plus will help stabilize the shoulder--with a pully or tubing-punch straight out, then push out a bit farther, like a karate punch.
If you do staggered pull ups, get both sides equally, even if one side is weaker. Do as many as you can on the BAD side, and the same amount on the good. It'll catch up.