Worst things first: That won't run AH3 well as such. The graphics aren't powerful enough.
The good things: The processor should be capable to run most games including AH, both 2 and 3. There's enough RAM for most basic purposes included gaming. The hard drive is a basic one with plenty of storage. Hopefully not a "green" one which would mean "slow".
If there's an empty pci-e slot on the motherboard, you could add a decent graphics card like a GeForce GTX 960 for about $200 or a better one for more, but you'd have to get a better power source to be able to run a good video card, too.
This is what I compiled from the cheapest (actually top 3) similar parts with pcpartpicker, adding a video card and a good power source. Add a keyboard and mouse according your budget and liking, also remove wifi if you don't need it:
PCPartPicker part list:
http://pcpartpicker.com/p/QbRnxrCPU: Intel Core i3-4160 3.6GHz Dual-Core Processor ($96.99 @ SuperBiiz)
Motherboard: Gigabyte GA-B85-HD3 ATX LGA1150 Motherboard ($52.90 @ SuperBiiz)
Memory: G.Skill Ripjaws X Series 8GB (2 x 4GB) DDR3-1600 Memory ($29.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($39.10 @ SuperBiiz)
Video Card: MSI GeForce GTX 960 2GB Video Card ($164.99 @ SuperBiiz)
Case: Thermaltake Versa H21 Window ATX Mid Tower Case ($24.99 @ Newegg)
Power Supply: SeaSonic 650W 80+ Gold Certified Semi-Modular ATX Power Supply ($59.99 @ Newegg)
Optical Drive: LG GH24NSC0B DVD/CD Writer ($11.99 @ Newegg)
Operating System: Microsoft Windows 7 Home Premium SP1 OEM (64-bit) ($119.98 @ OutletPC)
Wireless Network Adapter: Asus PCE-N10 802.11b/g/n PCI-Express x1 Wi-Fi Adapter ($6.91 @ SuperBiiz)
Total: $599.83Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-11-28 17:08 EST-0500