Which came first the pain in arm or the cough. If the cough, then
you could have pulled a muscle
in your shoulder region and are feeling some referred pain. If you're not allergic to it, you could try some Ibuprofen over the counter as directed to see if that helps it. That's some good stuff for any inflammation that you may have going on, just be sure to always eat something with it so you don't get ulcerations in your stomach.