Rhetorical, don't take it personally: If you don't know how to ride long distances, how "realistic" is your cross-country ride?
Be very afraid.
I'd have to say pretty realistic. Of the folks I on the Trans America a large percentage just decided to go and did it. Many would not have called themselves cyclists before the tour. Quite a few trained as they went if you can call that training. What most had in common was that they were stubborn goal oriented people, were generally fairly fit, and were experienced in other outdoor activities. I would think based on the folks that I know who successfully did a NT, TA, or ST that those qualities were way more important than how much they trained.
As far as the be "afraid comment". I have to say there isn't much to be afraid of on an XC tour. On tour, go out and ride however many miles you feel like each day and don't push yourself past your comfort zone. Pretty soon it is just the normal daily grind.
Obviously it is best to have some mileage in, but real training is not required. So yeah get some mileage in. It will help especially for the first 10 days to 2 weeks, but after a couple weeks you will be in the groove even if you weren't at the start.
Remember it is a vacation not a competitive event, so no need to train hard unless you just want to for it's own sake. The exception is if you are trying to do a fast crossing. In that case train like crazy.