How do the Alaskan Jews know when to begin and end Yom Kippur?

God goes by the principle of Mercy before Judgement, so if you failed to determine the "sun down moment event" with precision - and not to your fault - it does not matter. It is what "you do" what matters. The widow that could not afford a bull and rendered a dove had the same atonement the rich Lanlord had. On the other hand, there are watches, sun dials, radio, calendars, TV, etc...they are not living in Mars, you know... That is the same kind of question someone made - ions ago - when asked "(Gen. 3:1) Now the serpent was the most crafty of all the brutes on the earth, which the Lord God made, and the serpent said to the woman, Wherefore has God said, Eat not of every tree of the garden?" - in the Garden. Watch out for your heart. He reads them well.

Candlelighting times are usually given on a calender in any community. Figure the time zone and then go sounth to a normal sunrise sunset lattitude and use those times. :)

the sun still rises and still sets no matter where you are. i am sure they have rabbis who tell them what to do but if the sun sets at 2:00 am then yom kippur starts 2 am and then they finish fasting 3 am the next morning

i dont think there are any over there

It is an issue widely discussed amongst modern day Rabbis. There is no consensus on the matter. Some rabbis say to base the time off of the nearest locale with a "sunset time" other says Jerusalem time. Still others say Jews should not visit these places because of this issue.

Sundown is sundown - no matter where you are. For some Jews, it would mean a longer observance than for other Jews.

There are only 3,000 Jews located in Alaska (estimate by Wikipedia). And I'm very sure that there are not many people--much less any Jews, living in Northern Russia. Because Yom Kippur is the fasting of sun down to sun down, it should not make any difference at all. Everyday the sun rises and falls wherever you are on the Earth. As for other holidays, there are alternatives.