We are renting a car and want to board the ship as early as possible to enjoy the ship. But I screwed up and didn't get us fully checked in until last week. We are scheduled to embark at 1:30 pm - the earliest time slot left. Can we show up at the port early and try to board, or will we have to wait until our group is called at 1:30 no matter what?
What you have is a Port Arrival Time, not an embarkation time.
DCL started the PAT system to keep the crowds in the terminal at a more manageable level with the advent of the larger ships. You will receive your actual boarding number when you checkin at the port. I will point out that your boarding number is tied to your PAT.
The terminal opens at 11:00 (sometimes a bit earlier). Boarding generally starts about 11:45 (again, sometimes a bit earlier). At that time boarding #1 (Concierge) is announced, then #2 (Platinum CC members), then #3, 4, 5, etc. Once the number being announced for boarding catches up with the number being issued at checkin (generally in the 12:30-12:45 range), Open Boarding is announced. At this time anyone holding any boarding number may board.
You will receive the same boarding number, regardless of when you actually checkin. So, even if you are allowed into the terminal early to checkin, you will see others arriving after you (at their correct PAT) boarding before you do.
All that being said, the "sweet spot" for arriving at the terminal is usually between 12:30-1:00. It's usually after the Open boarding is announced (or close to it), and before the DCL busses from the WDW resorts arrive (with more people).