What is the easiest and cheapest way to be transferred from Genoa's Airport to the Cruises Port?
The easiest is certainly by taxi.
The cheapest sensible route would be by Volabus (€6 each, get off at Piazza Acquaverde in front of Genova Piazza Principe train station) and then walk (less than ten minutes).
Make a deal with the cab-driver for 5-6 € per person. ( works for 3 persons and up)
If you can get that deal, it will not be less than €7 per person for a minimum of three people. Normally, the deal only applies to train stations, not to the Cruise Port.