Rock climbing is not a particularly demanding aerobic activity. Muscle failure in rock climbing is generally due to getting ‘pumped,’ the condition when forearm muscles fill with blood and retain lactic acid (a by-product of muscles working), then they cramp, swell, and refuse to operate.
However, a certain level of aerobic fitness is certainly recommended for a rock climber. The approaches and descents from many climbs can be long and strenuous.
It is not unusual to hike 2 or 3 miles steeply uphill to reach the base of a climb. Schlepping a heavy pack of gear, ropes, and water adds to the effort. Descents can be tedious, long walks or scrambles down difficult terrain. For reasons of both added enjoyment and safety, rock climbers need to be in decent aerobic shape.
A minimum aerobic program for climbers would involve a couple of half-hour sessions per week combined with weekend activities. Suggested exercises include 30 minutes of running, biking, fast walking, or any other activity that keeps the heart rate up.
On the weekends (if you are not out climbing), go mountaineering, hiking, biking, cross-country skiing, or something similar. Probably the best advice is to create a lifestyle that includes different levels of aerobic activity that become a part of the day-to-day routine.
If your rock-climbing goals include bigger and more remote climbs that require long and steep walks in, then by all means step up your aerobic training accordingly.
Increase the frequency and level of training runs or rides. Take advantage of every opportunity to simulate the aerobic activity that you are preparing for; that is, carrying a heavy pack up a steep hill. Yes, humping a pack up a hot trail can be boring and miserable work.
Learn to pace yourself, stay refueled with water and snacks, and recover with rest breaks, and perhaps you will find it is not that bad. You may even find it perversely enjoyable.