Cruise lines often tell guests to 'not arrive until XYZ time' because they are trying to control crowding. They hope that at least a few will listen but don't expect everyone to comply. You can arrive earlier but to be honest you may have to wait for a little bit to board. If you arrive at 12:30 you may find there is no waiting line and you can check in and board in one continuous flow, which might be helpful to your father if he has any physical limitations. Personally, I always arrive around 11am or so because I can't wait to board!
How wonderful for you both! Celebrity is an excellent choice. You won't be totally surrounded by 'senior citizens'. A lot of people take advantage of cruises during this time of year because they usually are a bargain. The weeks between Thanksgiving and Christmas are a down time for cruises in the Caribbean historically. People are more concerned with shopping, and holiday preparations.
Cozumel is a pleasant island. It is quite commercial now as compared to the old days when it was really rustic. But, it is a safe port and the people are friendly and helpful. I like it.