Making a radio-controlled clock work from a local time signal
You could transmit on the standard frequency, or hack the clock to receive another.

How far do you want to transmit to this clock?

If only an inch or so, then you may be able transmit at such low power that you can drown out the national signal at the clock, and be so low power that you don't upset the neighbours, or the FCC. Any antenna would be near field at those frequencies, and so a well aligned nearly direct coupling to the clock's ferrite rod would end up broadcasting practically no EM radiation.

If you are trying to transmit metres, then that would be illegal in any jurisdiction. You would need to use a power that will upset your neighbours' timekeeping, and would bring down the wrath of the telecommunication authorities.

As for hacking a clock, it's a case of open one and see. Older clocks might be better, more likely to have a separate RF section with audio modulation out. If I was designing one these days, I might be inclined to do something clever with subsampling the RF directly after a bit of tuned amplification, and so bury the demodulator in silicon. If you can locate the modulation, then it should be a cinch to drive with the output of your 433MHz garage door opener. Test it with your Arduino directly first though.

Do you want the clock to actually be a radio wall clock, or look like one, or just look like a clock? With a display and an Arduino, you could write a clock, and equip it with your garage door opener receiver, and you're done.