Since it's a MOUSE PAD I don't think you necessarily need a lint-free cloth. You may even need a scrubber of some kind.
Again, while I think alcohol likely will work fine make sure to TEST in a corner of the pad.
*You really should have some Isopropyl Alcohol anyway as part of a first aid kit so it's a good idea to get. I may be incorrect but I think 70% solution is good both for that AND for using on the computer.