I wonder if they would be more accepting if you had a complete itinerary... or perhaps a Japanese or someone residing in Japan who could act as your 'guarantor.'
Would you be going as a part of a tour group, or soley alone?
If it were a part of an exchange, that would be different.
If you were so inclined you could just make friends with someone on the plane and have them act as your 'travelling companion'.
Are you running away, using Japan as a stopover, or just going for the fun of it?
Afterall youth often travel abroad to have someone meet them at the airport, who's to say that you arent having someone meet you?
One thing though, is that generally you will need to have addresses and or phone numbers of contacts in Japan.
Each time I have been there, I have found that they like to know my whereabouts... and Im ten years older than you.
If you are serious about this trip, there are things that you can do.
When I was younger than you, my mother was trying to ship me off to Europe for a few months of schooling, and then traveling solo to Holland and Switzerland. It was a fair distance from my home country.