The Cayman Islands are on Eastern Time, but do not observe Daylight Savings Time. So, between April and October, they are one hour behind the U.S. East Coast.
Ship's time will be the same as the local time of the next destination. So, ship's time will be for the U.S. East Coast upon departure and in Key West. After departure from Key West, probably during the night, ship's time will change to that of Grand Cayman (between April and October... between October and April, no change will be necessary.)
Incidentally, Cozumel is on Central Time, but <i><b>does</b></i> observice Daylight Savings Time, so there'll be a time change involved somewhere on the Western Cruise. The location of the change will be different depending on the time of the year.