Rome Fiumicino Airport to central Rome:

by dedicated 'Express' train or by regional train from the airport railway station to Rome Termini. Allow about 60 -75 minutes. 

Follow the signs to the station (there is a lift against the right hand wall as you approach it, opposite the escalator). Rather than queue for tickets use the automated free standing ticket machines, pressing the correct symbol for your language eg union jack. Very easy to use - credit or debit cards are best as they do not give change. These machines are common on all Italian railway stations.

 At Termini station follow the blue line alongside platform 24 - it's quite a long way to the concourse but it is level and you do not need to go to another floor.

Alternatively use the bus (cheaper but slower) that leaves from close to the Information office - you may be approached by the driver en route to the station.

  Rome Termini to Civitavecchia Cruise Port:

Trains leave about every 30 minutes from platform 25 or 26, a long but level hike from the concourse (where there are many automatic ticket machines - 4.50€ second class single or 7€ first class single. Eventually a very pleasant journey. On arrival it is a simple, level exit from the station - turn right (past the taxi rank) and trundle about 300m (all quite easy and pleasant - there are ramps at all pavement crossings) to the port where courtesy shuttle buses leave from immediately inside the gate (each marked with the name of the cruise ship they are going to). These buses are not allowed to pick up at the station but can drop returning passengers there... ! Allow about 90 - 100 minutes journey time from Termini.

Ciovitavecchia Cruise Port to Rome Fiumicino Airport:

Cheapest way is to reverse the above procedure and either return to Termini or change at an intermediate station then picking up a train to the airport.