I am arriving in Rome at 9:30am on October 8th and planning to take the Leonardo express to Termini. Do I need to purchase advance tickets to get to Naples? If so, what train time should I book?
You do not need to purchase advance tickets to go to Naples, and I would advise against purchasing anything in advance in the event your flight is delayed. You can certainly peruse the trenitalia.com website and see what trains are scheduled that day, their departure time, their travel times and their costs and do some pre-planning in that regard.
It's such a major line and October is outside peak tourist season that you really should wait. As Carrie pointed out, you don't actually know when you're flight is going to arrive nor how long it will take you to get out of the airport. Good for you for asking before you bought the ticket. Happy travels. (You do know all the standard warnings about Naples, right?)
The 11:08 Leonardo Express arrives at Termini at 11:40, giving you just enough time to catch the 12:00 EuroStar Italia train to Napoli Centrale. If you manage to catch the 10:38 Leonardo Express, the 12:00 ES train will still be the first one you can take to Naples.