WD-40 (or other similar type of spray-lubricant) works very well in removing old label residue. Spray it on the residue, allow it to sit for a minute, and it should come off with a paper towel very easily. If the residue is not thick sometimes you can just spray WD-40 onto the paper towel and rub the residue off with it. Good luck!
I’ve had great success with a product called Goo Gone. It can be found in Walmart, Kroger, just about any discount store and a lot of grocery stores. It doesn’t cost much and goes a long, long way.