"You look just like me when I was your age." NOT "You look just like I was at your age." "You look just like I did at your age" is correct and means the same thing.

Yes, you can. However in this case, "You look just like I did at your age" would make more sense. Another example might be, "You play sports just like I did at your age" or "You are tall just like I was at your age."