Purchasing train tickets from Rome to Naples?

I would like to purchase the train tickets ahead.

Be aware that discounted tickets can't be changed after the scheduled departure time of the train. Quite obviously, nobody gets full price tickets in advance.

If you want to purchase ahead to save some money, you are running the risk of wasting all your money. And no, high speed trains do not sell out.

ps Two train companies run high speed trains on the Rome-Naples-Salerno line: trenitalia.com and italotreno.it
Only Trenitalia serves the FCO airport station, but you can install both apps and make two separate purchases. Italotreno's walk-up fares are usually cheaper than Trenitalia's.

It's risky. If your plane is badly delayed you cannot get a refund on a discounted ticket purchased ahead. Actually if you arrive at the station 1 hour after the train is departed you cannot even get a refund for a ticket purchased at full fare.

The full fare, using high speed trains, is 62€ pp one way.
At most you can save 25€ or so on a discounted ticket. I'm not sure it is worth the risk.
In your shoes I would buy the ticket on the spot, when I get to the Fiumicino Aeroporto station, bite the bullet and pay the full price (62€). Seats don't run out.
The Rome-Naples line is also served by Regionale Veloce (speedy regional) trains, which are cheaper, but of course take twice as long.
Train timetable and pricing is below. Now it's too early to buy for October (in case you want to take that risk).
www.trenitalia.com

The discount is not that big. There is a connection to Naples from the airport every 30 minutes. Just buy when you have your luggage. You can download the Trenitalia app and buy it there, or from a vending machine, or from the ticket window. October is a lifetime away.

to be safe?

As above- purchase on arrival to FCO. Get the apps- both Trenitalia and ItaloTreno are easy to use
The LE from FCO to Termini can't sell out runs very 15 min or so and fast trains rarely sell out- there are multiple trains every hour..

Customs is just a matter of walking thru the correct door- no time at all.
Passport Control is what might take time and you cannot predict that or if flight will be on time or baggage comes quickly.

I would purchase tickets after you get thru Passport Control, while waiting for your baggage.

Certainly way too early to purchase now

Station names are
Roma Termini
Napoli Centrale

Thank you all so much.
I kept reading that the High Speed trains sell out fast, but was, also, very worried about flight being delayed.
I will go by your expertise, and buy them after I have my luggage.
Again, thanks for the help!

Can I ask where you are reading that fast trains sell out fast?

Definitely not true for Italy so maybe find better sources for your travel research

Maybe you have red that discounted tickets are capacity controlled and sell out fast. It trains sold out months in advance, railway companies would simply add cars.