The 7850 is a good upgrade, it takes you from a slow and outdated videocard to what is today an entry level gaming card. You could play any game out today at 720p on the 7850 and have a good experience, and many games will play well at 1080p too. You can't say that about the 57xx series.
The limiting factors are twofold. One, your CPU. Depending on which specific CPU you have, newer quad core oriented games may not run well on it no matter what card you buy. Two, motherboard compatibility. Depending on the motherboard bios your system uses, newer videocards may not work. The HD 7850 is guaranteed to work on your system as far as motherboard bios compatibility goes.