Oh, yeah that's like fractals that have infinite perimeter, but finite area (i.e. Koch snowflake).

You CAN "cover the surface" with paint, if the thickness of the paint is very very small. If you fill the horn with paint, you are effectively painting it, the thickness of the paint being the radius.

It stops seeming so paradoxical when you realize it's not just the surface area of the horn that's infinite, the length of the horn is also infinite.