Is that secondhand? That's a sweet deal either way, just make sure there's nothing wrong with it.
You'll probably be upgrading to a 1080p screen. In that case, the 5850 is a good performer at that resolution, allowing you to crank up the settings and add a bit of AA.
Your CPU will definitely bottleneck your GPU, even at high resolutions. If you don't want to upgrade yet, try overclocking instead. As for your PSU, as long as you have the requisite 2x6-pin PCI-e power connectors, it's OK. The TDP of the 5850 is 151 watts, and I doubt that the rest of your system consumes 350W.