Warren, the airport you'll be flying from is Ciampino.
I note the spelling only because it helps to get it right if googling around for information. :O)
Airlink looks like an economical plan. Looking at schedules for a random date (Sun, Aug. 7), I see that the early-morning combined journey of train + bus takes a little over 30 minutes. The first train departs at 6:21 on that Sunday, There are 2 earlier departures during weekdays (05:42 and 05:49) and journey times for those range from 36 minutes to 44 minutes, depending on what train your take. Tickets cost € 2.70.
You can purchase your tickets at Termini or pre-purchase them on the Trenitalia website. I would assume that if you buy a digital ticket through the app that you'll need to check in per this process before boarding, as is now customary on regionale trains.
https://www.trenitalia.com/en/purchase/digital-regional-ticket.html
If you purchase a paper ticket at Termini, you just validate it manually in one of the time-stamping machines before boarding.
But they're listing the prices for Terravision (Roma Termini to airport) bus tickets as € 6.00 and same for SIT buses. That's really inexpensive for a comfortable coach bus (we've taken airport buses in Rome and Milan) no transfer from train to bus, and no checking-in process other than your ticket may need manual validation when you board the bus. Skipping that transfer plus easy check-in would be worth the extra € 3.30 to me. Your larger luggage (if you have any) also goes under the bus so you don't have to keep eyes/hands on it on the train...which could be standing room only as regionales don't have seat assignments.
https://www.adr.it/web/aeroporti-di-roma-en/pax-cia-bus
I haven't looked at all of the journey times for SIT (they don't make as many runs) but the Terravision buses look to cover ground in about 40-45 minutes.