It will fit in the case, it should work with the system but... check to ensure you have the following connections available on your 500W PSU - two each PCIe 6pin (or 6+2pin or 8pin) connectors - they'll be needed to power the 560ti. You shouldn't have to upgrade anything (if the PSU has the connectors) besides the graphics card.
Be sure to wipe the 6670's graphics drivers from the computer before installing the 560ti to avoid driver conflict headaches