A little searching turned up:
Ring epimorphisms and C(X) by Michael Barr, W.D. Burgess and R. Raphael (article).
They consider this question for rings of the form of continuous functions on a topological space. They quote the following characterisation of epimorphisms in the category of commutative rings:
Proposition: A homomorphism f : A → B is an epimorphism if and only if for all b ∈ B there exist matrices C, D, E of sizes 1 × n, n × n, and n × 1 respectively, where (i) C and E have entries in B, (ii) D has entries in f(A), (iii) the entries of CD and of DE are elements of f(A) and (iv) b = CDE. (Such a triple is called a zig-zag for b.)
This seems a little more complicated than localisation, though I haven't checked the details.
They then go on to prove that
2.12: A subspace Y of a perfectly normal first countable space X induces an epimorphism if and only if it is locally closed.
If I understand all the terminology correctly, then this implies that
C([0,1],ℝ) → C((0,1),ℝ)
is an epimorphism.
There are plenty more references in that article, and it would be nice to have an actual zig-zag for this situation. But in the spirit of open-source mathematics, I thought I'd post this and see if someone (possibly me later on) can fill in the details.
Added Later: The example I gave: C([0,1],ℝ) → C((0,1),ℝ) is a localisation. It is obtained by inverting all functions in C([0,1],ℝ) which are zero only at the end-points. Given a function f ∈ C((0,1),ℝ), there will be a function g ∈ C([0,1],ℝ) which is non-zero apart from at 0 and 1 and which goes to 0 at 0 and 1 faster enough that the product g f also goes to 0 at the end-points. Then g f is (the restriction of something in) C([0,1],ℝ) and g becomes invertible in C((0,1),ℝ). So f = g-1 (g f) is in the specified localisation of C([0,1],ℝ).
Indeed, the Barr et. al. paper comments on the fact that in all the examples they consider (function rings), the zig-zag has length 1. I conjecture that if the zig-zags always have length 1 (for a particular function f: A → B), then B is formed by a localisation on A. A possibly stronger version of this conjecture would be that this is an if-and-only-if. In which case, finding a counter-example to Anton's conjecture would involve finding a case where there was a zig-zag of length 2. I suspect that a universal construction would be the best approach to finding one.
In the spirit of wiki-ness and only doing a little at a time, I'll leave this here.
Added Even Later: (Should I timestamp these? I know that the system does so, but is it useful to embed them in the edit?)
Here's one direction for my conjecture above.
If B = S-1A, then for b ∈ B, we have b = s-1a for some s ∈ S and a ∈ A. Then we put C = s-1, D = s, E = b = s-1 a. Then CD = 1, DE = a, D ∈ f(A), and CDE = b. So in a localisation, zig-zags have length 1.