Probably a simple solution, but can't think of much right now, so many possibilities. First, thought of just a transistor as a switch, off the clock speaker, but realized that it would switch at the same frequency as the alarm tone, which might work for a bigger speaker, but not the car horn. Maybe something like a sound switch, like the ones that you clap your hands to turn something on. Never built one myself, but have seen the topic come up several times.
It's tough to think, since there is a huge variety of alarm clocks. Mine sound like a pulsed tone, at a set frequency. If you knew the frequency, there was a cheap IC that could be set up with a few parts to give an on/off signal. Think it was LM567, have in seen it much in the past 20 years, might be out of production.