There is generally a battery that powers the clock and keeps it running even when there is no external power to the system normally one of the larger coin shaped ones. It's non-rechargable as it lasts for a long time. It's being constantly drained but it's only the smallest trickle.
If you can get to the internals you should be able to find and replace it, it's pretty obvious when you know to look for it. While you are in there you can check for corrosion/damage to the traces and components, give things a dust. The biggest issue is probably going to be an old battery, 20 years ago they were rather primative and though generally more replaceable than modern builds, getting a replacement might be harder.