For $500.00 or less it is really tough to build your own machine because Windows alone eats about $100.00 leaving very little to buy reasonable quality components.
You may want to consider looking at www.techbargains.com, on occasion I have seen Dell offering reasonably decent full systems for under $400.00 - whether they will play the games you mentioned well, I cannot say but, if the system has a PCIe 16 slot you could always purchase a video card later.
the A8-3850 chip w/ DDR3-1600 should get you about 35-45 fps at medium settings in most modern games at 1600x900.
the red bar in this chart is 1680x1050 performance at DDR3-1600. 1600x900 should be similar