The best way to get to Rome from the port is by train according everything I have read.
There are regularly scheduled trains from the port city to Rome all day long (Europe is big on train service).
If you try the train look into getting reserved tickets (much more expensive) but otherwise with so many cruisers you may find trains filling up before you can get on. I have personally never done this, but I know a lot of people have done it.
You can also get a car - but it will not be cheap. But it is cheaper is you have two or more people than paying the "per person" price for transport by Royal Caribbean buses.