The Naples airport is Capodichino. From the Airport the city center is easily reached by taxi or by bus. From the Airport to the Astronomical Observatory of Capodimonte the most effective way is a taxi (~15-20€).
In case you fly to Rome, you can take a high-speed train which will get you to Naples in about 1 hour 10 minutes. There is a train connecting the Rome Fiumicino airport to the Rome railway station every 30 min. and from there trains to Naples every hour.