Fly in Rome then train immediately to Naples -timing and what if flight is late

It is best to buy tickets upon arrival for the very reason your flight could be late. If you want to take the risk of having to throw away discount tickets and buy new, full price ones, give yourself 4 hours to get to Termini. That builds in a buffer and gives you time to have lunch if you are time. (I have done this. Got 39 Euro tickets so worth a try. It worked. )

The price for the Leonardo Express train to Rome Termini is alsways the same, 14 EUR. The Freccia trains to Naples are 46 EUR walk up price, a ticket bought 60 day or more in advance are 16.90 EUR. So to save 29 EUR are you willing to take the chance? Just buy when you get there and save the worry.

I would agree with Laurel. Four hours would be the min. We have sailed through immigration in 45 minutes - usually about an hour or so, and then one Saturday it was nearly three hours. You take the Leonard Express to Termini - twice an hour and takes about 30 minutes. So allow an hour should you just miss it. You can buy both the LE ticket and your ticket to Naples at the ticket office at the airport. HOWEVER, be absolutely certain you validate (time stamp) your LE ticket before getting on the train. It is an immediate, pay on the spot, no mercy, fine of 50 euro each if you don't.

Can I buy these tickets on my cellphone when there or must be on paper?

If you end up stuck at Termini for a while before your train departs or if you want to buy some food for the train trip, go to the Mercato Centrale in the station. There are different areas, like kiosks but permanent, with different categories of food, and everything we tried was good. And fast. Lots of people have their suitcases with them, so you don't feel silly the way you might having luggage in a restaurant.

Speaking of kiosks, that's exactly where the OP can buy tickets, if they don't want to wait in a queue at the ticket window at Termini. Pretty self-explanatory, and don't let any weirdos try to 'help' you work the machine! Take your time, get your tickets, grab a panini to go while you're waiting, and enjoy the hour-plus ride to Napoli.

Whew! Never experienced Termini with jet lag. I love it, but it's a real rogue's gallery there.

Both the Leonardo Express and your ticket to Naples can be purchased at the airport train station. No need to wait until you reach Termini.

Definitely buy your tickets at the airport; nothing like a lenghty delay in your arrival to screw up those nonrefundable tickets.

When I am at the airport can I buy my tickets on my phone ?

You can and the locals do all the time.