All of your destinations/transits are easily reached by train
Trenitalia
https://www.trenitalia.com/content/tcom/en.html
Roma Termini- Assisi
You will take the Leonardo Express from FCO to Termini then catch train to Assisi- this will be a regional train.
Check the schedule now so you have an idea of your choices, no point purchasing in advance as you don't know if your flight will be delayed or how long Passport Control will take. Since this will be a regional train - no need to purchase in advance as the price never changes and they can't sell out. Be sure to validate your ticket.
You could also download the Trenitalia app and purchase your ticket once you have landed and are thru Passport Control-
or
Just buy your ticket at the airport station (window or kiosk) for the LE and your train to Assisi
There are trains that require no change, some require a change in Foligno.
The journey will take about 2 hours.
A private driver for that distance would be pricey- no idea for sure but I'd guess 400 euros? (the train from Termini is 13.80 euros, LE is 15 euros per person)
Once you arrive at Assisi station you can grab a taxi or the bus up to the town. (The station is well below the town)
Take a taxi to the airport- fixed rate 55 euros. Have your hotel call for one that am- it shouldn't take more than 30-40 minutes to get there. Very easy. Be in a taxi by 6:15 and you will be fine.
OR
If you are staying near Termini- take the Leonardo Express from Termini to airport. I think it runs every 15 min.
What time is your flight?