You'll be fine. Start with good tires. Wife & I each carry a tube & patch kit, I carry a mini pump and we each have a co2 system. Stan's Dart if we're on big tires and tubeless. A 4" piece cut from an old tire is our boot if needed and flatly slips inside a seat pack. We've done the GAP, C&O, Katy and a bunch of self supported gravel tours. We generally only get flats when in a town or on a road shoulder. Even then, a flat is a 10 minute time out and sometimes in a beautiful place - if the latter I opt for the pump and save co2 for another day. I have truly good memories of flats in Vermont and Montana
Enjoy your trip and don't worry.