This page should help you identify it. If you can measure the thread size, you should be good.
Edison screw - Wikipedia, the free encyclopedia
You just need a 6V E10 bulb (~0.6W).
The bulb has a resistance of V/I=60 Ohm. A 24V/0.35A bulb is a 68 Ohm bulb, and would draw just under 0.1A, so it wouldn't damage the clock, not sure if it would be bright enough. It'd probably last forever though, considering how far under wattage it would be.