You are severely limited by your budget to perform the upgrade and to be worth it.
Yes, i3 4170 is a better CPU than A8 7600, but it is not the only component that matters. Are you using the integrated graphics from that APU or a separate discrete graphics card? If integrated, then your current A8 is actually a better solution for gaming than i3 4170, which has a weaker integrated graphics. And if you plan on adding a discrete graphics card, it certainly won't be cheap.
Lastly, i3 4170 is not the latest gen, and when purchasing new you should get i3 6100 (which also contains a better integrated graphics than your A8 and would be better in every aspect). But that would require you to invest into DDR4 memory.
Your call.