I haven't airpumped while wearing a cockring, but I've worn one several times using a
hydropump. Basically, just pay attention and you'll be fine. For one a cockring inside a
hydropump can help with
gaiter bite or directing expansion more to the shaft. Or perhaps you want to expand your glans more, then you'd place a cockring behind your glans and pump.
Caution is adviced strongly. I have not had any issues the times I've worn a cockring inside the
hydropump, but it's not something I'd suggest a beginner to do. (Not calling anyone beginner in this thread right now, but overall!)
So, if you feel no ill effects, go ahead, but my experience only stands for
hydropumping.
Anyhow, I recall that a way to use an air pump is basically to put on a cockring, then pump and get erection and wham and bang. (Thank you m'am.) More catered towards older men I think, but that's one way of utilizing an airpump.