Can a Phenom II 965 @ 4.0 GHz handle a 7970?

It will not bottleneck much and it is already overclocked so it would be fine.if you have budget then you can get fx 6300 as you will get more performance.

It will bottleneck it although you should still be able to play games at 60+ fps in most games. In my opinion the difference between a 965 and a FX 6300 does not warrant an upgrade. I advise you to wait for the AMD steamroller CPU's (should be out by the end of they year) because the indications are that the chip will have even better multitasking performance and single threaded performance will also improve dramatically to the point where it will compete with Intel Haswell.

It depends on your resolution if its under 1920 x 1080 then the CPU will be a large bottleneck @ 2560 x 1440 then not much at all. It will also depend on the game, like Metro will nearly always be limited by the GPU.