Hi brownie77 and welcome to Cruisemates! We've found that a long sleeved shirt, a tie and nice slacks are fine for Formal Night on Princess. If you want to rent a Tux on board, you'll be fine wearing that, too.
Many men wear a sport's coat or a suit, but it's not necessary. Alaska cruise's seem to be a little less formal than some other's.
Have a wonderful cruise,