For people who live in the extreme reaches of the North above the arctic circle, how are they to use the moon to determine their Sabbath when they have either 24 hours of sun or 24 hours of dark?

Answer: That is a great question, but the answer is quite simple.  I myself live in a Northern climate.  In the summertime, the sun is up well before we are awake and sets after we go to bed.  On Preparation Day, we are ready for Sabbath when we go to bed, knowing that it will already be Sabbath when we wake up.  Likewise, that evening, it is still Sabbath when we go to bed.  In the wintertime, we remain consistent.  In the interests of "guarding the edges of the Sabbath" we are fully ready for Sabbath when we go to bed.  We do not slip in a quick shower or a load of laundry just because we "can" since it is still dark outside and the sun has not yet come up.  During the long winter evenings, the Sabbath is technically over, but we desire to retain the blessing of the Sabbath and want it to linger.  We do not plan a big party or the week's laundry for the evening following the Sabbath as soon as it gets dark.  The same principle holds true for both extremes: 24 hours of light or 24 hours of dark, regardless of whether one uses the Gregorian, luni-solar, or the Chinese calendar: on the evening of Preparation Day, you go to bed all ready for Sabbath, knowing that when you awake, it will already be Sabbath.  You continue to keep Sabbath until you go to bed that night.