Just talk to her on Facebook. You won't know until you try. The worst that will happen is that she won't be interested. You've got nothing to lose because it's not like you're going to have her if you don't do anything either.
Don't worry about the boarding school issue. You haven't even been in a relationship where you've felt strongly about somebody so how can you know how good or bad being apart will be? My husband and I were in a long distance relationship for years when we were dating, and not seeing each other often really wasn't as bad as a lot of people think.
Anyway, right now you've just spent one evening with her, so you're getting ahead of yourself worrying about how the relationship would be. You can cross that bridge when you get to it. At this point your next step is to see if you can continue getting to know her, so you can find out if she's really as incredible as she seems. Good luck!
“This planet has - or rather had - a problem, which was this: most of the people living on it were unhappy for pretty much of the time. Many solutions were suggested for this problem, but most of these were largely concerned with the movement of small green pieces of paper, which was odd because on the whole it wasn't the small green pieces of paper that were unhappy.” ― Douglas Adams, The Hitchhiker's Guide to the Galaxy