If it's a quality PSU, sure, but 300W would be really pushing it. I'm not even sure if a 300W PSU meets the minimum requirements. You'll want at least 20A or better on the 12V rail. If your PSU doesn't (this should be on the sticker on the side of the PSU itself), then buy something 400W or better from Antec, Enermax, Tagan, OCZ, Sparkle/FSP/Fortron, Seasonic, or Silverstone.
And, yeah, so long as you have an AGP card, yes. PCI Express is not compatible with PCI slots, much less AGP.
The 6800GS is the AGP 6800 variant, and will require direct power from the power supply.