(1) Your interpretation is basically correct. "To show him the mirror" means to do something to him that he often does to you, so that he can see the error on his ways from the other side, as the receiver of his acts rather than the doer. Therefore your sentence (slightly corrected for idiom)
He's become very arrogant. I told him how he used to come to me to borrow money when he was a student, to show him the mirror.
is not quite correct, because merely telling him anything is not showing him the mirror. Although people do say the phrase the way you have said it, it is not the proper way to show the mirror. This, on the other hand, is:
He's become very arrogant; he's always too busy doing important things to say hello. But he's a mooch. So, just to show him the mirror, when he came by and asked to borrow some money, I told him, just as he always does, "as IF I had the time!" -- and slammed the door in his face.
(2) I see mirror daily in the morning when I shave means I am reminded of my true nature every day. The idea about the mirror is that you see yourself in it as you really are, not as you would like others to see you.