rather than staying at an airport hotel, I would suggest taking the Leonardo Express train into Rome and to stay at a hotel there. Rome has so much much to offer that I wouldn't waste my time staying out near the airport.
From downtown Rome (Roma Termini station), there are regular trains to Civitavecchia from where your cruise is likely to depart. See here
how easy it is to get from Rome to Civitavecchia by train (or vice versa).