We had taken the train before to Naples then got Private car ride to Sorento we booked in advance. . Is this doable for a day in Naples?
It sounds like you have plenty of experience. Later you say that Pompeii is not the objective this time. As long as you have a low-priced, non-changeable, non-refundable Freciarossa train ticket (i.e. really fast), it's a reasonable plan. It takes almost as long to get to Villa d'Este or Hadrian's Villa from Rome, by Metro and bus or train.
But you imply there's more than Naples to the trip. The Amalfi Coast (which does not include Sorrento) is a bit more remote, even by private car. Herculaneum and Oplontis are little closer to Naples than Pompeii and Sorrento, and are both accessible by Circumvesuviana. Have you considered Caserta Palace as well? Don't overlook the Farnese Marbles if you go to the Naples Archaeological Museum.
You don't give the month of travel, but note that we waited over an hour for a seat at a mid-famous pizza place in Naples. I mention that also because the Naples museum can be stuffy and crowded. I'm spoiled by their traveling fresco show that came through NYC this year, but the Naples display layout can be poorly lit in places and hard to see, like flat cases in front of wall displays.