What methods are there to travel from Siena to Monteriggioni?

Answer 1

You can take a bus. There are regular connections during the day from Siena to Monteriggioni (however the service is suspended or slowed at evening-nigth).
You can also take a train from Siena to Castellina Scalo and at Castellina you can hop on the shuttle bus going to Monteriggioni.
Also this service is bette working on day time.
In the night you can take a taxi (though very expensive).

Answer 2

It's easily reachable driving with the speedway Florence-Siena from the homonym exit. Monteriggioni is about 15km. from Siena.