It's true, the clocks is finicky...it might not be your bulb. My clock light has been off and on for the last 3 years or so, the bulb isn't burned out, there must be a loose connection somewhere in the line/socket, cause it goes off and on every once in a while. When it goes out, I remove the clock and wiggle the connectors, tap the clock, plug it in and it usally comes back on.

And yes, there is a bulb to replace, it's very small, but you'll figure it out.