I am not sure. We don't know yet. Is the earth flat or a globe?
It's a globe. If it weren't, you will very quickly find out, but the cruise company already knows it is.
Determining position it's interesting. We can't rely on GPS. We will have to use the old methods plus compasses to determine which way we travel. Any ideas?
Can you find someone to teach you how to do celestial navigation using a sextant, then practice, practice, practice? You will need to have accurate time to get accurate longitude, but, fortunately you have several options there.
You can use a GPS receiver for time, ignoring its reported position. This is probably easiest, but, y'know, GPS...
There are several shortwave stations that transmit little more than very accurate time signals, so get a general coverage shortwave receiver and learn the transmitter frequencies of several of these; you'll figure out which band to listen for at what time of day after a while. Downside: these stations are always operated by governments. Maybe compare some from different countries against each other - I'm familiar with two: CHU (Canada) and WWV/WWVH (USA), but there are bound to be others, and since they transmit on several different shortwave bands at the same time, all the time, they have worldwide reach much of the day.
Even cheap digital watches these days are very accurate, so get three or more and keep a log of how well they agree with each other, and with civil time, over an extended period before leaving, so you'll know which are running fast or slow, and by how much. Continue logging each against the others (and, if you have SW radio time, or GPS time, against that) during the voyage. Do not reset any of them while testing or during the voyage, ever, just note and keep track of what corrections need to be applied to each, and use the corrected times when calculating your position fix. Realistically, three Timexes, Seikos, or Casios (or one of each) will probably be within a few seconds of each other, and the correct time, after several months, so the whole correction table thing may be moot, but using three or more will let you know if one is developing a problem. This may be the most "independent" way, but could be the most involved (or not, if the watches are reliable and accurate enough).
It would no doubt be entertaining to see how well your celestial positions match GPS. You'll probably have lots of time on your hands.
This could be a lot of fun, though, and you'd learn an arcane skill. You can amaze your friends!