We just stayed at the Rome Airport Hilton. If you have access to the Executive Lounge they do have lots of variety for breakfast, not just coffee and pastries. . There is also a giant buffett breakfast downstairs. If you are Hilton Gold or Diamond, both are free. I think they both open for breakfast as early as 6am. So you could also just have a later breakfast, just before you leave.
As for lunch or snacks, maybe the easiest thing would be to pick something up the night you arrive at the airport, before you make your way to the hotel. To get to the hotel, you are going to be waking through the airport- and probably taking the walkway to the hotel. You might want to buy some snacks a long the way to save for your trip.
As for takeaway options at the hotel, I'm sure you could also find something in one of the restaurants. You could call the hotel in advance and ask them. Otherwise, pick something up on the way back through the airport to the train.
Lastly, depending on what class of service you have to Florence, you might get a free drink and snack (it's not really free because it's a more expensive tickets). However, the trains usually have options to buy food or drink.