I'd have to read the ingredients in GooGone before using for this. You dont want to strip the paint off the walls! I would probably try a hydrogen peroxide all purpose cleaner first, and use a dish sponge that has two sides: one side is foamy sponge and the other is a scrubber. I buy these at either Dollar General or Dollar Tree. $1 for a pack of maybe 8. These foamy little sponges are wonderful because they can do so many things in cleaning. I even use them to get pet hairs off furniture!
Let us know what you try and how it comes out.