You'll probably see some improvement, but as sizzling said it isn't going to work well at the resolutions that card is designed to run. If your only problem is the 2GB frame buffer, you should see a huge jump, but that's only going to happen in very niche situtations.
While I wouldn't recommend investing more in a dead platform, an FX-8320 might be a good stopgap if you're budget is tight. The FX-6300 is essentially a 3 core CPU with hyper-threading and the cores themselves are not very fast to begin with. The FX-8320 is slightly faster out of the box and works as essentially a quad core, which most newer titles are becoming dependent on. While the FX-8350 and above are considerably better, they are going for over a hundred while the...