VRAM is simply how much information the video card can store to work with, it says nothing about actual speed.
It all comes down to effective speed. Let's use AMD Phenom II x4 965 Quad Core 3.4 ghz, a solid AM3 cpu that i would recommend for you due to its good single core performance that comes in handy for both java games and 3d games and good overclocking potential especially if it's the Black Edition which you could push up to as much as 4 ghz with a standard aftermarket heatsink
heavily depends on your motherboards VRM though, you might not be able to achieve these speeds...Anyway, this chip is able to be paired with as much as a gtx 960 with only a little bit of bottlenecking in 3d games from the chips side of things. essentially the cpu can just barely not keep up with it. Now if we compare the 960 with the r7 240 which is another 2gb card but is in the 60$ range we see that in tests it's only about
25% as fast. Essentially it's too slow to keep up and will be a bad pairing in 3d games. This says nothing for 2d, however, as they will be roughly equal performers.
Also the reason nvidia cards tend to have less VRAM than AMD but still perform as well comes down to having a much higher cache. I don't want to get into the advantages of that though so ill just say they are both good.
but more VRAM is good to have if you're at higher resolutions are are working with more than one monitor, that's for sure. I don't know what your monitor situation is though, and really the r7 240 is fine if you truly intend to not play gpu intensive games at all on your new cpu.