PCIe x16 slot power has to be able to support up to 75w by specification.
A 6 pin connector has to be able to support 75w by specification.
An 8 pin connector has to be able to support 150w by specification.
So a graphics card with two 8 pin connectors could theoretically have the ability to draw up to 375w, but would typically draw considerably less than that, but could also technically spike to slightly more than that.
Buying an HD 7770, NOW, 7 years after it was manufactured, would be a really poor idea. Any HD 7770 has likely already seen the majority of it's useful life expectancy pass by. Unless you could get the card for less than fifty bucks, I'd pass.
What the listed capacity of ANY power supply is, is not nearly as important as what the MODEL of the PSU is, so that it's quality can be gauged. There are FAR more power supplies out there that claim to be 500w units that can't reliably sustain 350w, then there are ones that either can sustain their listed capacity or even exceed it. Which means, there are a whole LOT of crappy power supplies out there and just because it SAYS 500w, does not by any stretch of the imagination mean that it actually can. Even a fair number of well known brands have significant numbers of models that are very poor quality.
So, knowing what MODEL that unit is you are looking at would be a very good idea, and would allow us to help you to decide whether it's a complete waste of your money or a pretty safe bet, one way or the other.
What country are you in?
How much are you able to spend on a graphics card and power supply?
What EXACTLY is this system going to be primarily used FOR?
For future reference, I prefer to use THIS list of recommendations when it comes to how much capacity a given graphics card should be supported by when choosing a PSU. Yes, in some cases the numbers are slightly more than what is strictly necessary, but that is by design, to offer some measure of headroom for the sake of safety and the health of you PSU, graphics card and motherboard. You won't make a mistake if you use this for assessing how much capacity you should be looking for. But keep in mind, that is IF you are also going after a quality power supply. Not a mediocre or cheap unit.
http://www.realhardtechx.com/index_archivos/Page362.htm