Fly to Naples, Capodichino Airport: the B&B Casa del Sole is not far from here, only 8 km and takes just 15 minutes by taxi. Ortherwise you can take the ALIBUS, check timetable and prices on this website http://www.anm.it/Upload/RES/PDF/ORARI/C03%2012%20apodichino.pdf
It will drop you in Piazza Garibaldi (train station), from where you can easily take the metro.
Line 2 (the only one you can take from there), direction Pozzuoli. From Garibaldi to Cavour (1 stop). Then change metro line 1 (take the underground passage). Direction Piscinola, from Museo station to Materdei (1 stop).

If you arrive or land in Rome, you can take the train from Termini (2 hours with an intercity train that costs more or less 20 euros, or 1 hour if you take a Freccia Rossa that is quite expensive, more or less 50 euros, but you can find many offers) Check on www.trenitalia.it
You can also check the new train timetable and prices, its name is Italo and from Rome Ostiense or Tiburtina arrives in Naples in about 1 hour and half www.italotreno.it

Cheap airlines: www.easyjet.it, www.airitaly.it