Hairspray will take permanent marker off clothing. I spray it on a Q-tip then rub it on the stain, but you can also probably spray it directly on the fabric.
I've found that Sharpie is not all that permanent on plastics like water bottles, so I think it would come off the cooler fairly easily. Goo Gone, petroleum distillates, mineral spirits (paint thinner), nail polish remover - any kind of organic solvent should do it. Just don't leave it on too long and be sure to wash it off well - solvents can eat plastic.