If your armpit pain is in fact another nasty side effect of your period, the good news is that it is short lived. However, the bad news is that there’s nothing you can really do to treat it. The only thing you can do is minimize the discomfort, the same way you would with any other type of menstrual pain. For example, try going braless for a few days, or while you sleep, as restricting the area may increase pain in your breasts while on your menstrual cycle, and thus also increase the pain in your armpits. Also, many have found success with different relaxation techniques, such as lightly massaging the area!
That said, if your pain persists, does not only occur during menstruation, or if you have other symptoms, like swelling or the presence of a painful lump, you should see your doctor. If you see a rash or other signs of skin problems under your arm, then see a dermatologist. It might not hurt to see a doctor if the pain is a temporary problem either, as getting a medical diagnosis can also ease some anxiety.