Shoulder/arm pain after playing badminton. is this normal?

No, it cannot be ---: normal. Could be reasonable if you played after a long time, and not having done any warm up exercises. Suggest: do regular stretching exercises for your shoulder muscles, about 3 times a day & take some Advil or aleve (naproxen). A hot pack alternating with an Ice pack for 15 minutes each, about twice a day, would be helpful. If no improvement after 7-10 days, then see your PCP/Orthopod. Good Luck.