Couple of problems. Modern systems pull their power from the 12V rail. You have (12V * 13A) 156W available. The TDP of the 8800GT is around 110W, and I'm pretty sure your CPU will want more then 46W, and that leaves nothing for your motherboard, drives, etc.
You have other problems like a P4 isn't strong enough to handle an 8800GT, and less then 2GBs of ram. Probably best to start working on a whole new system.
I have looked at the specs and all I gather is at least a 450w psu. Its a generic brand from what I gather, all it says is powercase on the unit itself
Also, I have found that my specs will support it, but will be a little bottleneck which does not concern me too much.
My current gpu is dieing and this is going to be a cheap ($50) replacement until I can afford a new computer.
It is incrediably low for a "500W" PSU, but it is the info provided. I wouldn't be surprised to learn its really 18A as the 8 could be looking like a 3. 12V * 18A = 216W which is getting better. But 110W GPU + 50W for board and drives leaves him with only around 56W for his CPU. I'm pretty sure the P4s use more then that. My answer would still be no.
Other then the PSU I think he's ok. I'm pretty sure he has one 6pin PCIe plug, and a PCIe 16x slot seeing as he is using a 7950GT right now. Just has the power issue and the P4 issue.
The reason I want to replace the 7950gt is due to it overheating. I've cleaned the entire case, fan, card, got a decent airflow and fans both intaking and exhausting, yet everything but the gpu is at a cool temp. At idle i get about 62c and when i play games like tf2 or borderlands i get terrible tearing.
That being said, could this psu be a cause for the gpu fan to be spinning too slowly and hence underpowering it?
Would it be worthwile purchasing a good quality 450w psu in this case?
Yes the P4 will "support" the 8800GT, but I think you will find that your frame rates won't change much. The P4 is pretty slow by todays standards and won't be able to supply data to the 8800GT fast enough. I seriously doubt it will be any faster then your 7950GT.
I'm not sure the PSU is the cause for any slow downs. Should be replaced because its a low end one, but other then that I don't see it as "bad" from a pure performance point of view.