Training one-arm pull-ups is a popular activity for climbers because it translates well to bouldering and lead climbing, enabling you to power through hard moves. Chances are, if you can do one-arm pull-ups, you can remove upper-body strength from the list of reasons for why you are not sending your project.
Generally, to complete a proper one-arm pull-up, you must be able to do at least ten neutral grip bodyweight pull-ups, three or more pull-ups with 160%+ body weight, and a few controlled one-arm pull-up negatives. It will benefit you to focus on these requirements in this order during your progression to one-arms.
Body Weight Pull-ups
The best place to start is by testing your max-reps for bodyweight pull-ups. If you are already able to do ten or more reps in a set, move on to weighted pull-ups. Otherwise, start training for one-arms by doing more bodyweight pull-ups. Do not attempt to complete your max-reps every training session. A good workout for someone who can do a maximum of six pull-ups is to complete nine sets of three reps, alternating grip types each set.
It is challenging to do an actual one-arm pull-up because, whether you use a bar or rings, your shoulder will naturally rotate to the chin-up or neutral-grip position. So, as you train to complete ten bodyweight pull-ups in a row, you should also train chin-ups and neutral-grip pull-ups equally. These will build more bicep strength, which is necessary for one-arms.
After you can do 10-20 bodyweight pull-ups, chin-ups, and neutral-grip pull-ups, start training weighted pull-ups. Begin with only 10-20 lbs and increase the added weight each week so that you can only complete 5-7 reps during each set. Continue to train all grip types during this phase. As the added weight gets heavier (around 30% of your body weight), you should start doing sets with only 3-5 reps each. Also, note that this is strength, not endurance, training, so get as much rest as you need between sets.
Your goal with weighted pull-ups, in the context of one-arm pull-ups, should be to complete three reps at around 160% body weight. So, if you weigh 150 lbs, your goal should be to do a few reps with 90 pounds added. That is a significant amount of weight, so expect the weighted pull-ups progression to take around six to eight weeks.
One-arm Pull-up Specific Exercises
It is likely that after achieving two-arm pull-ups and chin-ups with 160% body weight, that you can complete a single one-arm. However, to perform a proper one (without kipping, using momentum, and lowering the right way), you should train one-arms more directly first. Two good ways to do this are through assisted one-arms and one-arm negatives. These exercises will help you develop the strength that is explicitly required for the one-arm movement.
Once you can do three one-arm negatives in a row with each arm, you should be able to complete a single one-arm pull-up with good form. Don't stop here! Continue training this movement and progress toward doing multiple one-arms and weighted one-arms. At this level, pull strength cannot get in your way of achieving your climbing goals.