If this information does not pan out to be accurate or convenient...many airports offer individual restrooms just outside the main ones, you could de-robe and do a wash up in sink, towel off and put on fresh clothing...I am sure that you are not the only one who will have done this! :)
Good luck perking up before driving on jetlag...check the forum for driving in Italy tips re. traffic violations, parking etc. Enjoy!
From http://www.sleepinginairports.net/europe/rome.htm
Airport Lounges
If you are an economy class traveller, you can enter these lounges at Rome Fiumicino airport either by purchasing a lounge pass or through a membership program. Visit the individual lounge page for more details (ie. facilities, hours, pass and membership information):
I Mosaici - Location: Terminal 3, Airside, 1st Floor, Gate D5 (Schengen Area). Showers are available. Hours: 8:30AM - 6:30PM (Daily). Lounge Access: Pay at the Door (30 EUR) • Buy a lounge pass (£25.00)
Le Anfore Business Class Lounge - Location: Terminal 3, Airside, Satellite Area, opposite Gates C30 and C31 (Non-Schengen Flights). Showers are available. Hours: 6:15AM - 11:45PM (Daily). Lounge Access: Pay at the Door (30 EUR) • Buy a lounge pass (£25.00)
Avia Lounge - Location: Terminal 3, Airside, 2nd Floor (escalator near Gates C and D). Hours: 5:30AM - 8:30AM (Daily). Lounge Access: Buy a lounge pass (£25.00)
Alitalia Lounges - There a 5 Alitalia lounges (three in Terminal 1 and two in Terminal 3). Rate: Free for Ulisse members. MilleMiglia members pay EUR 30.
» Buy a Lounge Pass