The cheapest way to get to and from the airport into Genoa itself is by the 'Volabus' bus service, which links the airport with Principe and Brignole train stations, stopping on route at Cornigliano and Sampierdarena train stations (useful if you plan to take a regional train to Arenzano, Savona, or points W of Genoa) and in Piazza De Ferrari.  The ticket costs € 6 and includes a 60 minutes journey on the Amt network ( buses, metro, public elevators and funicular railways), in the same day.  Further information in English http://www.amt.genova.it/rete_e_orari...  

In case you plan to arrive in Genoa by train, the city has two major train stations: Principe and Brignole, as well as many smaller neighborhood stations.  All trains stop at both.  Principe is W of the center, about 1 km from the cruise and ferry terminal, and walking distance from the historic section of the city and the Aquarium. Brignole is E of the center, near the bus terminal, the fairgrounds, and not far from the soccer stadium and the seaside promenade (Corso Italia).  There are hotels around both major train stations.