Italy Trip in October

It is only okay if you buy a type of ticket where the terms say that it is refundable or exchangeable in a case where your flight is delayed that you missed the 11:08 train. These are the most expensive of tickets.

You are not really reserving a ticket you are buying a ticket with an assigned seat.

My last trip to Italy a month ago I bought a ticket before I left at a great price 30 euros Rome to Naples for two people.
I lost those non refundable non exchangeable tickets because my flight never made it there and then had to buy them again at 3 times the price when I finally got to Rome.

I would not. The problem is that if your flight is late, the ticket is worthless. It is extremely unlikely that the trains would sell out, so just buy once you have passed through Passport Control. Make sure you know the next trains after the one you hope to get so you are ready with a backup plan that might have you arriving later than you desire.

Do you have checked luggage? Actually, I hope you are travelling really light because that's a heck of a journey you have ahead of you - getting through the airport, then a 5 hour train ride (or longer), with changes in Rome and again in La Spezia.
If all goes well, you should be able to get the 11.08, if not, then resign yourselves to the 13.08 (via Termini, not Tiburtina). But there are half a dozen common reasons you could miss the train - a weather delay (never happens at O'Hare, right?), a mechanical delay, plane congestion at Fiumicino - a holding pattern to land, or sitting on the tarmac waiting for a slot, a long line at passport control . . . okay that's only 5, but they've all happened to me somewhere sometime. There are more potential issues if you check a bag.

When you purchase your ticket, be sure to ask for the shortest trip time. I see some routes with Frecciabianca trains (faster) vs, the Regionale (slower because they make many stops). When you’re riding with jet lag, those slower trains seem even slower!