There are several large gardens in Naples, including:
Real Bosco di Capodimonte, located up on the hill where the Capodimonte art museum is located inside the former royal palace. It was formerly a hunting preserve for King Charles of Bourbon and today features an English garden.
Parco Virgiliano, up on another hill overlooking the city and built into terraces, on a clear day you can see Sorrento, Capri, Ischia and Procida from here.
Villa Floridiana, on yet another hill this park features lots of walking paths and large lawns, as well as a ceramics museum.,
Parco Urbano dei Camaldoli, a little further out from downtown and on another hill, this one the highest in the area, this is a more "natural" park, not as groomed as the others but with amazing views.
Villa Comunale, this is the parkland that runs along the waterfront in Chiaia, quite close to the port.
There are more, but these are the main ones that I can think of. All of these are easily reached from the port using public transportation.