In reply to Andrew Mallinson:
Search on here for tendonitis etc, there are some really good threads with lots of useful info. Pushups etc can help a lot but I think apart from taking it easy for a while to let the inflammation subside, the #1 thing to do is examine the way you climb and find what it is you have to change to stop this from occurring - could be technique, frequency, intensity, warmup/cooldown, type of training and very likely a combination of most of these.
I have never tried armbands myself so can't really comment, the only thing I would say though is that while many things might help speed up your recovery, reduce the pain etc, don't lose sight of the fact that you need to find the root cause of the problem go make your climbing injury-free on an ongoing basis. Good luck!