Is this a known problem with this generation Airport Express?
Is there a fix?
If you haven't yet, I would recommend that you perform a "factory default" reset on your Express to see if this helps resolve the problem.
Even though the Airport Express' original warranty has expired, if you have an Apple computer (iMac, MacBook Air, etc) that is within warranty period, including AppleCare, Apple will cover the Airport Express. This is why it pays to get AppleCare on Macs. Having said that, you may want to try the reset as suggested by Tesserax. Another thing that came to mind is power: is the Airport Express plugged into a power strip? Sometimes older surge protecters can become unstable and cause "flickers" in power delivery. I've had this happen to me before.