I don't wear a watch, but on a cruise, I set my phone to manually keep time. The signals you get will mess you up and the cell towers may not project the same time as "ship time." Sometimes, it'll be really weird. Passing from Canada to southern Alaska, we once had all of our phones displaying a time that was 1:15 late, and a date that was several years ago. I kid you not.
Bring some large and small plastic baggies. They can be put to all sorts of uses. In extremely hot places (such as jungle excursions in Cozumel), if you get out of an air conditioned van or bus and take out your camera or camcorder, the cool hardware hits the hot wet air and it may draw in some heavy condensation that will be slow to evaporate. If your camera is inside a tightly seal baggie, wait five minutes for it to warm up, and when you take it out, you'll have no condensation. They are great to put your phone in if you go out on a jet ski, too.
Bring singles and small bills for tipping the luggage handlers on the way on and off the boat.
Be five to ten minutes late for all meals. Arriving early or on time usually just means standing in a long line. Arrive a little bit late and you'll be taken care of right away.
On Disney ships, don't plan on gambling. There is none. Don't get too excited about the night life. It is pretty subdued. Do see the shows, even if you aren't normally into cruise ship shows. The Disney ones are vastly superior to other lines' shows.