Continuity is important not because of its inverse-image-ness, but because the definition corresponds to the geometric notion that it's intending to capture. A continuous function "takes close things to close things". This geometric intuition precedes the notion of open sets, and even the epsilon-delta definition of continuity.

I think the same sort of thing is true of functions in general. The asymmetry is there as a consequence of the idea that the definition is intending to capture; functions weren't originally thought of as a special class of relations, they were thought of as "rules" for manipulating numbers, and the idea of f(x) being unique and f${}^{-1}$(x) not being unique is the only natural way to capture the idea of a "rule" in a more general context.

I don't know that the notions of monomorphisms and epimorphisms really "correct" for the asymmetry, but I don't think it's something that ought to be corrected anyway. Monomorphisms of sets are the same as injective functions, and epimorphisms of sets are the same as surjective functions, and that's the way it should be. In categories like Ring where, say, epimorphisms aren't always surjective, it's because the non-surjective epimorphisms, in some sense, are "surjective as far as ring maps are concerned"; for example, the inclusion of $\mathbb{Z}$ into $\mathbb{Q}$ is epi because a map out of $\mathbb{Q}$ is determined by what it does to $\mathbb{Z}$. I don't think this has much to do with the asymmetry you're describing.