For a restaurant you must go to Ippogrifo Ristorante, located at Via R. Gestro, Tel 010 59 27 64
http://www.ristoranteippogrifo.it. My cousins who live in Genoa brought me there for lunch and I thought the food was so great we went back the next day for dinner. FYI, it is less then a 10 minute cab ride from The Hotel Bristol