Plan on riding less mileage for the first few weeks. One, you are getting used to riding day after day and your muscles need to adjust, and two, the Appalachians have many steep hills which you'll encounter those first weeks. You can make up the mileage later in the trip. And remember to plan a rest day every 7-10 days.
The cheapest way to go is to camp and cook your own meals. Assuming you have all the necessary gear before you leave, a budget of $35-$50 day should be enough to pay for camping fees and groceries and still stash a bit away for unforeseen expenses like repairs or gear replacement. Going dirtbag-style, it’s possible to travel on $10/day or less.