I vaguely recall my primary school teacher reading us a book featuring a unicorn. The unicorn's horn had healing powers and the main villain was someone who had the tip of a unicorn's horn stuck in his sternum because he jerked away when the unicorn was healing him - as a result he's always in pain but always healing. The story focuses on a young "teenage" unicorn (who is actually like 90) and he goes on an adventure with a boy. I think they go to an alternate "fantasy" world. At one point the young unicorn touches a spring with its horn to purify it.

This was read to me as a kid in the 90s, so it was probably written in the 80s or earlier.

I can't remember the name, and all my googling just comes up with The Last Unicorn over and over! halp.

Edit: More info! Probably before 1996. Could potentially be an NZ or Australian novel but not necessarily. Also, I think the boy was in a normal modern society before going on his magic adventure (think Narnia).