SSince you are flying in a day early, you might also consider staying at one of the hotels or motels nearer the port.
From the airport to the port is about a 47 minute drive.
However, there are hotels that are under 10 minutes from the port, some of which offer a free shuttle service from the hotel to the pier.
You might want to consider a hotel in eithr Cocoa Beach, or if you want the very closest one, take a look at the Radisson at the Port. It's about 5 minutes from the pier. And they have a shuttle.
You can see how close the Radisson at the Port is to the ships using their webcam.
It's a controllable cam. If someone else is already contrtolling it, their time will be up in 2 minute4s, then it's your turn. Use the scroll bar to move the picture, and you'll soon see the ship looming over the roof of the hotel, IF it's a day when cruise ships are in port. That would be any day EXCEPt Tuesday or Wednesday. They generally arrive around 6 AM and are gone by 5PM, so you need to look during the day.
Several of the car rental companies do one way rentals from the airport to the port, too. You can drop off your car at the rental agency near the port, and they will take you to the port from there. (They will probably also drop you at a local hotel, but I'm not sure about that.)
In general, the area surrounding the port is much a more pleasant and relaxing atmosphere than the area around the airport. (in my opinion, but I live here, so I'm biased.)
Out in Brevard county, where the port is, there is a beach (and the ocean!), rivers, lots of green, waterfront eateries, great seafood, little shops, and things like that. Much more relaxing than the area around the airport, with is mostly wall to wall traffic, concrete, freeways, parking lots and chain restaurants.