A crew on an elegant sailing boat has left an island located 13°S
latitude a few days ago, always sailing almost exactly east. It is now
Tuesday, March 22, 2005. One hour later, it is Monday, March 21, 2005.
They continue their eastbound course and again a few dozen
miles/kilometers/nautical miles later, they arrive at another island.
The total trip at sea is about 772 miles/1242 km/671 nautical miles.
What is the name of the island they arrive at?

I don't have the reference tools at hand to confirm it, but my guess
is that the answer is either Upolu or Pago Pago in the Samoa Islands.

My bet got to Apia 
since the day changed you're in long 180
if you zoom out on mapquest

http://www.mapquest.com/maps/map.adp?searchtype=address&formtype=latlong&latlongtype=decimal&latitude=-13&longitude=180

at around 772 miles/1242 km/671 nautical East of the red star... there is APIA

good luck

http://www.mapsouthpacific.com/samoa/index.html

Apia is the city the island is on of the 4 islands on the map
http://www.mapsouthpacific.com/samoa/index.html

Savai'i, Western Samoa.

The island is Tonga, situated at 19d.35.281m South and 174d.17.442m
East. Your figure of 13deg. South seems to be to the north of the
islands. One hour later, travelling east, they have passed the
Internation Date Line [I.D.L.][180deg.E/W] [geographical, not
national]putting them in the previous day. Estimating the sailing
speed at 8 knots or 8 nautical miles per hour, they started their
journey 8 nautical miles west if the I.D.L [ 179deg.52min East]. This
would comply with crossing the I.D.L. one hour later, then the day
changed. To continue their eastbound course for twelve nautical miles
would place them at 179deg.48min West. One nautical mile equals
approx. one minute of longitude, being close to the equator..  Your
trip of 671 nautical miles equals 10.01 degrees/ So from your starting
point at approx. 179deg52 min East , travelling 671 n.m. places you at
170deg.o7 min. West. According to my G.P.S., there is no island at
that position, but if there is it is pretty small. And having looked
my George Philip World Atlas, it shows the position of Tonga as 19deg
50min south by 174deg. 30min "West". This I think is a mistake as the
longtitude should be East. The next airfield to this is Honiara and
that is 159deg. 56min EAST. I stand by ready to be corrected.

I am correcting myself..Apia is at 13deg.50min south by 171deg. 50min.
West. I was not too far out.