I think your plan to stay in Naples is correct, as a convenient base for day trips.
Rushing on the arrival day to Capri, though, seems a little of stretch, either for the 2 hours time from the airport to ferry and the little 4 hours on the islands.
Capri is a magic place, providing you stay well far from the crowd and tour groups, and the best way to do so is to overnight.
Once the last ferry to Naples departs, the place turns in a more quite, relaxing and enjoyable pace.
A late night stroll, through narrow streets and alleys, reveals a whole new unexpected charm.
Bear in mind that Capri is not a "beach" place, unlike Bali, Koh Samui etc.
I would categorize it more a see and be seen spot.

If what above is not an option, you will have the time from the port to climb up via the funicular to the Piazzetta (main little square), walking past the hotel Quisisana, heading left for the pleasant via Tragara, which leads to the terrace overlooking the Faraglioni, local landmarks.

A bite and a drink on the way back to the ferry, will seal your glimpse of Capri.