It could be easier than you think. A thimbwheel switch can set the alarm in BCD digits and use cascaded comparitors to determine when it's equal, now you will have at least a pulse when the comparitor is true.
The question is what to do with the pulse? Buzz forever? Buzz for 30 minutes? Snooze?
Now, I'm not sure what kind of clock this is, but the DIP switch seems to allow you to set individual digits. If it's a digital clock, there is no AM/PM indicator unless it uses 24 hr time.