How much will a 8350 boost performance in games like Crysis 3 and Arma 3 coming from a phenom 965

My current gpu is a gtx 560 non ti if that helps

not much, upgrading your GPU will make far more difference than any CPU upgrade

the GPU upgrade will do much more, the Phenom II X4 965 BE at stock clock has about half the computing performance of an overclocked FX8350, but you'll only be able to measure that, it won't show such a difference in user experience, in fact, you'll notice only the better memory access bandwidth of the FX, which does make a difference, but as the Phenom is a full quad core and the FX8350 is a quad-module rather than a quad-core, the performance doesn not scale 2:1 at all.

The Phenom as a gaming CPU is nowhere near bottlenecking even a 7970, when you upgrade your GPU, you'll still get the full performance boost of that upgrade, it will not be capped by the CPU. Probably there will be some bottlenecking with a dual 7970 crossfire or a 7990, but those aren't that efficient anyways in terms of upgraded performance for the extra cost.