I would imagine that they'll ferry the parts, mechanics and any tools they need to Yakutat on one of their cargo flights or a combi, and fix it there. As a boat captain with two engines I really didn't like leaving port on one screw. It's one thing to finish a trip if one engine fails, another to start a trip down one engine. I can't imagine many 737 pilots would want to take off to do an over water (or over mountain range, depending) flight down an engine.
Google Map "Yakutat Airport" and they have a nice picture of the Alaska Airlines terminal so you can see what they are working with.
Perhaps Flys320s can weigh in on his thoughts.