Can a person have a blood pressure reading of 120/80 and still have a partially blocked artery in his arm?

Yes. Blockages in the arteries providing blood to the arms are common. Most often these blockages don't cause any symptoms. It is good to be aware of them as they can give you a false low blood pressure. A simple way to test this is to check the arm pressure in both arms. The higher pressure is the accurate one. A 20mmhg difference is considered significant. An ultrasound can give a more accurate diag.
Yes! What is important is the blood pressure difference between both arms. If one arm 20 points lower there may be some blockage.