While it is easier to put all the info in from the website, you have to put in personal info, passport and credit card before selecting your port arrival time. On the app you can go directly to that and fill in the other info later from the website. Earliest time will be about 11am. Checkin generally starts at midnight of 45 days before cruise where the ship will be. So it began for 15 April Fort Lauderdale sailing at midnight east coast time last night (28 Feb).
Some ports check your arrival time when you arrive at the terminal, some don't. Some only care if there is a line forming.