I don't remember exactly how many cores boost, but the FX8300 will boost to 4.2GHz as required (it'll be 2-4 cores, I'm pretty sure).....
I don't believe CSGO is going to benefit from >4 cores at a higher clock.... at least not by much.
Regardless, there's no real teardowns of that system that I can find...... cooler height will be a major consideration. I'd suggest measuring the height clearance available (board PCB to side panel) to determine what room you have for a cooler.
"Budget" options like a 212 EVO, or Cryorig H7 would be decent options for the money - but I wouldn't look to invest much more than the required $25-$35 into an FX based platform.