Ok first of all, I see your money is tight so we shouldn't be telling him to wait for zen or go intel. We also aren't building him a beast computer, we are making his computer do what he wants which is gaming. Battlefield 4 at higher settings very much likes more multi threading performance, I know his gpu is fine for now. He just needs a cpu for cheap. I would go with a fx 8350 to be under his $200 budget. Make sure you have 8gb of ram at minimum.
Summary:
Q: "I was wondering that if I upgrade my CPU to a FX-8350, would it increase my FPS drastically"
Answer: Yes, this is likely why that 280x that should play bf4 just fine is struggling.
Q: "would I also need to upgrade my motherboard as well"
Answer: No, as long as you bio's supports an 8350, which I'm sure it does. Don't plan on overclocking.