Please don't drive jetlagged.
Look into taking the Flixbus from the Rome airport to Siena --- it looks like there is one at 10:45am and takes 3 hours and costs about $15 each. Drops you off in Piazza Gramsci.
We have not taken this direct Flixbus, but we have taken the Flixbus from Roma Tiburtina station to the Siena train station and then got up the hill to the old center by a series of escalators --- you could instead take a taxi to your hotel.
Flixbus has a nice website and an app. Their buses are comfortable coach buses with a toilet and wifi. Your other option is to take a train to Florence and then a bus to Siena but that will take almost two hours longer and doesn't leave until early afternoon.
I do see the temptation to rent a car and drive, but besides the jetlag, a rental car will have its own hassles and expenses. Anyway, look into the Flixbus as an option.
If you don't already have plane tickets to Rome, have you looked into flying to Florence instead? Then taking the bus to Siena.