While most hotels provide free shuttles to the Port of Miami, I have found that the car rental companies (Avis, Dollar, Thrifty) are more flexible.
We just came back from Carnival Triumph (Aug 26 - Sep 2). We stayed in the Red Roof Inn (Miami Airport) the night before the cruise. Red Roof provides free shuttle once a day (unless too many people going) to the pier but only at 1pm. If you take that shuttle, you will be at the pier around 1:30pm at the earliest. However, it will be extremely long lineup.
We instead returned our car to Thrifty (2 minutes walk from Red Roof) at 10:30am and got on its shuttle in about 20 minutes. The driver Daniel was great. He even showed us where he would pick us up a week later. We were at the pier around 11:30pm. We got through the security and Carnival line quickly and were enjoying our first lunch on board around 12:15pm.