I've been flying for about 25 years now. My very first flight was when I was 10 and all by myself. It went very well! The flight attendants helped me to get on-board and stow my carry on. I don't even remember who I sat near.
Now, I fly a lot. I love flying, on a big jet with lots of people, or a small plane all by myself. I don't worry so much about people around me, though sometimes the inconsolable baby with ear pressure can get annoying (but it isn't their fault, and I know that).
Anyway, tell the flight attendant that you are by yourself. They will be very happy to help you. Make sure you bring a magazine or two, book, iPod, or other entertainment with you in your carry-on. Anything electronic can't be used until the plane is over 10,000 feet (they'll tell you over the PA system when it is OK). You can read or just look out the window until then.
You may want to keep your carry-on under the seat, this way you can get to any entertainment you bring with.
If the person you are next to is weirding you out, just ask the flight attendant if you can move. If the flight isn't full, that is not usually a problem.
Anyway, have a great flight. I think you'll have a good one, and many more after this!