best solution - rent a car and drive yourself
Ciao Maggie,
Given the time you get into the Rome airport, the best solution for your family is to rent a car and drive yourself to Cortona.
But take into consideration the time it takes you to pick up your luggage and go through customs, then pick up the car rental and get going. The drive from the airport to Cortona is then about 2.5 hours --- you'd be getting to Cortona way past midnight, or more like 1am at the least.
Consider this:
1. is your accommodation a hotel with 24 hour reception? if it is a villa, it is unlikely that the owners or rental representative will stay up waiting for your arrival to hand you the keys. this is something you also have to consider whether you drive yourself or you hire a driver.
2. where are you flying in from? will you be rested enough to get behind the wheel and drive in a foreign country at that hour?
Wouldn't it be better to spend the first night somewhere near the Rome airport and drive to Cortona the following morning?
The drive itself should not be too daunting as you will not be going into Rome - you'll be taking the ring road (circovalazione) that surrounds Rome without ever entering the city. This is as long as you stay outside of Rome that first night.
The drive takes you on major toll roads that are pretty easy to drive on --- my suggestion is to drive when you are well rested and not stressing to get somewhere the following day. A GPS in the car would also be really helpful.
I've mapped the directions on Google which you can see here -
http://maps.google.com/maps?f=d&sou...2.003387,12.260742&spn=0.678626,1.579285&z=10
Hope these suggestions are useful!!