UPDATE:
I recommend an i5-4460 or i5-4590. I just checked PCPartpicker (USA) and they don't cost much more. Only about a $10 to $30 difference over the FX-8350.
The i5-4590 Turbo is 3.7GHz and the i5-4460 is 3.4GHz. For the price difference I'd get the i5-4590.
Here's a benchmark for the FX-8350 (again most games are worse and some match the i5):
http://www.tomshardware.com/reviews/fx-8350-vishera-review,3328-14.html
*4th gen Intel is about 10% faster than 3rd gen. Thus a 3GHz i5-4xxx is comparable to a i5-3xxx at 3.3GHz.
Other:
You can even avoid a custom CPU cooler and use the stock one if your budget is currently tight. I'd still recommend the CM Hyper 212 EVO for about $30.
You also need a better CPU cooler for the FX-8350...