I would STRONGLY advise against this. (And I'm someone who would ordinarily consider such a thing --- but having been to Barrow...) The "night" (its all night) will cost you no less than $700 for air from ANC and hotel. If you have any AS miles and feel the urge to go, then use your miles. It is a very expensive flight. You will have to stay at the Top of the World hotel, which is "acceptable" (about old Motel 6 standard, but for Barrow, fine). You will not be able to be outside for more than a minute or 2 at a time. Food is marginal and expensive. If you really want a winter Alaska experience, I might suggest Chena Hot Springs, where you have an excellent chance of Aurora viewing and can hottub at the same time (only 60 mi from FAI on an improved road).