Personally, I think the most important skill for a fiction writer to have that doesn't directly have to do with writing is empathy. Being able to really put yourself into the perspective of someone who is not you and perhaps not very much like you at all is crucial to writing well-rounded characters. (And yes, to some extent that's an innate ability that people have or don't have, but it's also a skill that can be developed. Even if you're good at it.)

-J