The GT 730 64 bit DDR5 will safely run on a much smaller PSU than even 300W. It really is not a very high wattage card. Only around 25W maxed out.
Yes, the PSU recommendations are always based on the entire system. Usually, the gfx card would be the most demanding component. In this case it is not. So, you are good with your 450W PSU. The 720 uses less wattage than your 8800 Gt.
Now for the bad news...
The GT 730 is not going to be any faster than your 8800 GT. You probably won't see much of an improvement unless the extra 512MB of faster VRAM helps somehow. That is why I was hesitant to recommend a better card on a $60 budget.