One of the higher end AMD APU's should fit your needs and budget, though you may not get Ultra settings in BF4. And if you are not looking to overclock, then just use the stock cooler that comes with the APU.
Hard to recommend a cheap cooler though without more info, like what case you are using. A good cheap cooler is the Coolermaster Hyper 212 Evo. Ive seen it in the $20-35 price range.