Naples can be easily reached by plane
http://www.aeroportodinapoli.it/on-departure/airlines
or railway (it is connected with Rome or Milan by high-speed trains)
http://https://www.trenitalia.com/en.html
Both the airport and the railway stations are close to the Observatory. The best choice is to take a taxi (15-20 min). Typical costs are about 15-20 euros.
There are plenty of things to do, visit and eat in Naples and its surroundings, due to the millenary history of the city.
To learn more about Naples and its attractions please visit:
In Capodimonte:
Museo di Capodimonte http://www.museocapodimonte.beniculturali.it/information-en/
Catacombe http://www.catacombedinapoli.it/en#
In the city center:
https://www.museoarcheologiconapoli.it/en/
https://www.napolisotterranea.org/en/naples-underground/
https://www.laneapolissotterrata.it/en/home-ing/
Pompei:
http://pompeiisites.org/en/visiting-info/
https://www.trenitalia.com/it/treni_regionali/campania/pompei-link.html