Can you upgrade the graphics processor & RAM on the Lenovo H500 desktop?

No you can't upgrade the memory. 4GB is the max and this computer is a Celeron. If you can please look at the Intel i3, i5 or i7 processor. It would not be worth it for you to upgrade the video card when you can't upgrade the memory and the processor is not very fast.

The only processor upgrades you can do are ones that support socket FCBGA1170. While the ram can't be upgraded past 4gb, you can add in a better video card, but you'll be limited on what you can buy by the weak power supply in the PC. Your best bet would be to sell this tired beast and buy a new PC.