Taxi to the port.
The city us close but not close enough to, the port is at the southern tip of the city but in a Long Island off if the city and so you need a taxi, taxis are regulated and easy to get at anytime and also any size.
When you say get there early, I say not before 10 am, the terminal will not open till exactly 10.30 am, you can queue outside, part of it is sheltered, but it's likely to be very very hit there, do not go too early, and it doesn't take long to get there.
When it opens there are plenty of check in desks in an open hanger style building, not a lot there. Check in is fast, but you will not start to go on until 11.30 am, boarding us by boarding card issued at check in, so actually little point turning up early, you set yourself up for an hours wait.
In last years a walkway grudge from the terminal building to the ship was added, in the last you entered by deck 2, now it's like port Canaveral you walk to the atrium. Therefore it's fast and by boarding card you get on very quick.
I have used pre booked taxis and on demand taxis, booked by my hotel, the on demand taxis were always better, they were there quick and pre booked let yes down more than once.
Public transport is great around the city and to and from the airport but due to where the port terminal is over a long bridge then a long pier, you need a taxi.
We stayed at The Hotel Jazz a great hotel, modern rooms, rooftop pool, close to the central Hub of the city. From there is was ten minutes to the ship.
DCL hotels are often out if the city to the east side and further from the centre and port.