Hi Patricia. “Should?” I can’t really answer that since the “should” of anything depends a lot on the facts and circumstances of what’s happening. Let me answer by saying that under “normal circumstances” the arms don’t get noticeably tight when cycling, but again a lot depends on several factors (e.g. your body position when cycling; do you have any contributing physical/health condition that could/might cause this etc.). I would highly recommend that you make an appointment to see your physician and have him/her give you an examination to determine if there is any issue that needs to be addressed. I would do this NOW before continuing to cycle. Then and only then would I look into other equipment-type issues.
I hope that this helps.