Yes, even though it sounds counterintuitive to go north to the airport (assuming you will be in the centre of Florence (Firenze) just before you leave) and then a little further north to get onto the Autostrada heading south, it actually works very well.
The airport is quite close to the centre, easily reachable, and the Autostrada swoops gently all around the city as it makes its way south. Is is absolutely trivial to get onto it from the airport, whereas it is a little complicated and risky of fines (especially as you will be getting used to the probably manual car) if you go from the centre.