Pcie 1/2/3 etc is irrelevant, they are forward and backward compatible. And... no card sold today will be impacted much by running on a slower interface anyway.
Your 300w psu should be good enough to run a card as strong as a 7750 that is powered entirely by the pcie slot.
Possibly, the new GTX640 will be ok.
Otherwise, you are looking at a psu upgrade also.
If you are looking at a really strong card, or an older one, check the length of the card, and how much space you have in your case for it. Some can be 11" long.
Given that the system is running a Core I3 2120 processor, I would expect it to have a PCI-Ex16 2.0 (or 2.1) slot. This means it could accept any PCI-Ex16 graphics card, as there are no compatibility issues with this version of PCI Express.
The system has a 300W power supply, and I would expect you could swap that out, easily enough with a better power supply.