Just noticed you have a family of four. So these are the figures.
The full base train fare from Napoli Centrale to Rama Termini for 4 adults in standard class is 45€ each. However there are discounts for children, families, and also for advance purchases. So the private car to Naples plus train will cost you no more than 280€. With discounted tickets it would be less than 200€. Travel time total, door to door, is less than 3.5 hours (90 min drive to Naples+train wait time, let's say 30 min to be super safe+70 min train ride+20 min at most estimated taxi ride from Rome station to your accommodations).
If you drive all the way from Positano to Rome with a private driver, it is a 4 hour drive total, door to door, so slightly longer. The price of 450€-500€ is about right.