Absolutely no problem, as we see in this video of "Man of Steel" and "Batman v. Superman: Dawn of Justice" stars Henry Cavill and Amy Adams taking part in the ALS ice bucket challenge.
Adams was challenged by partner Darren Le Gallo, and in turn, challenged her siblings. Cavill was challenged by numerous people, and didn't issue one in return.
Both actors are dressed in their costumes as Superman and Lois Lane, and gamely sit through one round of frigid water getting dumped on their heads. But then the crew decides to put Cavill to the test and pour increasing amounts of ice water on him!
Watch Superman get very, very cold: