Is it normal to have a sore arm after giving blood?

Yes. Yes, it is normal to have a sore arm. I would suggest over the counter pain medicine such as tylenol (acetaminophen) or ibuprofen. A warm compress may help as well. If the soreness does not improve in a couple of days, talk to your doctor.
Sometimes. People can often have a sore arm if they have a "hematoma" which is a little leakage of blood that occurs outside the vein after blood is drawn. The soreness typically will go away after a few days. If it does not or if you have any numbness or tingling in your fingers, you shoud see your doctor.