For the most part, cruises leaving in the late afternoon from U.S. ports will actually start boarding around noon. Sometimes 11am or so can be had too. We usually show up at the pier between 11am and noon and start our cruise. No need to wait until 1pm.
This past December we sailed on the Freedom out of Miami and arrived at the pier around 11am. We were on board by 11:30. There weren't many people in the terminal so check in was fast and easy. Between 1-2 we noticed the ship starting to get busy.
I've been at the terminal by 11am had to wait until afte 1pm more than once. Depends on how the disembarkation goes. Most of the time I'm onboard by 12:30.