How to get to Matera from Naples?

Not sure it can be done. The best connection I can find, using a pretended date a few days from now:

Leave Napoli Centrale at 8.07am
Change in Caserta, train to Bari Centrale at 9.15am, arrives 12.04am
On a side of Bari Centrale there is the little station of Ferrovie Appulo-Lucane (tracks are at a raised level), leave Bari FAL at 12.32am, arrive in Matera at 2.10pm

On Trenitalia you can find a couple of alternative choices, leave Napoli for Salerno, switch to a Freccialink (minibus run by Trenitalia) that allows you to be in Matera in four hours from Naples. But Freccialinks run only on afternoon.

I have tried to find connections via Potenza but I was not able to find a working bus timetable between Potenza and Matera.

Matera is the only provincial city in Italy not reached by Trenitalia trains; only by buses and by the FAL train line from Bari.

There are a couple of possibilities, not sure if either will work for your time constraints:

1. Fast train Napoli to Salerno, transfer to FrecciaLink to Matera. Last summer this link only ran twice a day.

2. Early train Napoli to Bari Centrale, go across the street to the smaller train station and buy your ticket for the regional train (different company, not Trenitalia). This train runs more frequently than the FrecciaLink, takes about an hour.

I have done both. The FrecciaLink is a minibus that you board outside the Salerno train station.

There's a Marino bus departing Naples at 9:10 and arriving in Matera at 13:45. Naples to Matera involves 4+ hours of travel, so finding a connection that arrives before noon could be a challenge. Another option would be to take an evening train part-way, say to Foggia (about 3 hours), and then you could get to Matera much earlier the next day. That option would still leave you a full morning and afternoon to explore Naples before departing on an evening train (18:07 might work best).