There are left luggage facilities at Naples Central station.
After being closed for quite a while, the LLO at the station seems to be open again.... but it would do no harm to phone the day before and check! See reply 8 here for the number...
Alternatively, there's often a seasonal baggage store in operation at the harbour - in the lefthand end of a wooden "Portakabin" immediately "ashore" from where the hydrofoils land at Beverello... however I've not seen news of whether that's running this year - either already, or at all!
But the rail station is probably more convenient, as you're likely to pass near there on the way to the airport.... just be sure to allow plenty of time to retrieve your bags, as they can be quite busy (especially if the new owners have cut back on the number of attendants!).
Indeed it might be simpler (and a deal cheaper) to come to Naples on the Circumvesuviana, and head straight to the one there?
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