Never done one privately to Capri, we found what the ship offering was actually okay.
I guess you are docking in Naples, advice #1 is be so aware of that port and be on your toes, as it is the robbing capital of Italy.
If you really want to to do Capri yourself, get off the ship and walk out the port building, turn left to the public water buses that go to Capri around every 45 minutes or so.
Get off in Capri and explain to the local taxi drivers what you want to see and they will do it,,,,,the cost split between a family of 5, its not that expensive.
But ask if they speak English and listen to the reply
and then ask how much for them to show you the island for X time. But please explain you must be back to the ship by Y time. Plus you want a driver who can explain what you are looking at.
As I say I have not done this, but we know people who have and they enjoyed it.