Approximation algorithm for Feedback Arc Set
Kann's online compendium of NPO problems is a good place to start. Feedback Arc Set (the "Directed part is redundant when you use "arc") is:


APX-hard,
Approximable within $\mathcal{O}(\log n \log \log n)$ (where $n$ is the number of vertices).


The problem is also fixed-parameter tractable1, so it might make more sense to solve the problem exactly, rather than use what looks like a bad approximation algorithm. (Or as Pål points out below, the running time is a bit... unpleasant, so maybe not.)

Notes

1 - JACM publication, Razgon &amp; Sullivan's preprint, Chen, Liu and Lu's preprint - the problem was independently solved and the results combined to one publication