You can definitely get from Rome to Florence taking a fast train, either with Trenitalia or Italo.
The Castello where you're staying at is about 20 km outside of Florence, their site says the closest train station is at Montelupo Fiorentino but that is still 10 km from the castle....
From Florence you can also take a SITA bus heading to Certaldo or Montespertoli, getting off at either Baccaiano or Montagnana. Or you can contact them and ask them to arrange a private taxi to pick you up either in Florence or at the Montelupo Fiorentino train station, whatever you feel better/easier for you.
Since you will be staying the countryside, you might consider renting a car to get around, if you're not just planning to stay there the whole time. You can read some tips here: http://www.discovertuscany.com/tuscany-car-rental/