See the fuller question for far more. But for a quick answer on Barrow specifically, here's the Star-K:
For example, in Barrow located at the northern tip of Alaska, on December 1 the sun does not rise. However, at 1:15 p.m. there is the most sunlight of the day 30 (theoretical chatzos hayom [=noon]). Therefore, one may daven
Shachris and perform daytime mitzvos between 10:40 a.m. and 1:15 p.m.31 (during these pre-dawn light conditions) and daven Mincha at 1:45 p.m. (1/2 hour after chatzos during the post sunset conditions). Shabbos ends at 4:23 p.m. when the stars come out.
I don't see them addressing when Shabbos starts, specifically. Possibly best to avoid working as early as 1:16PM on Friday, I'd assume. In which case it would be best not to light shabbos candles; by the time it's sufficiently "Friday afternoon" to light shabbos candles, it may already be sunset and prohibited to do so.) So probably light candles, without saying a bracha, before 1:16PM. I don't think their article addressed this one ... anyone?
This one's a bit easier. When there's no good sunset, we treat sunset as 6PM standard time. Hence, on standard time, the ideal time for lighting candles is 5:42PM, before 6:00PM standard time (likely 6:42PM--7:00PM with daylight saving), but if you wanted you could light candles as early as 4:45PM standard time.
bring along an almanac from home; if you came from somewhere where sunset in the winter is before 6PM (would have to be South of the equator if this happens in the summer, I think), then start shabbos at that earlier sunset time of your hometown.