Just have to figure out now the circuit and how to program it to send the impulse every second and the hour impulse every hour.
I don't think you'll need any of that. I think it uses a one minute pulse, not a one second pulse, and I think the one hour pulse is only for resynchronisation of clocks that have got out of step.
Looking at the photo I can see a partially hidden gear wheel which looks like it is probably part of a traditional "motion work" (ie the 12:1 gearing between the hour and minute hands). I am almost certain that if you apply a one minute impulse to the solenoid the clock will just work and the hour and minute hands will stay in the right relationship.
I would approach this by stripping out the PCB and any other bits and bobs and just reducing it to a solenoid. I don't agree with
@electrofix when he says the impulse driver (what is that?) looks too large. The size of it doesn't necessarily relate to the power required; it's easier to make efficient magnetic circuits if the parts are of decent size. Obviously I might be wrong about that; the only proper approach is to strip it down to the solenoid and then just try it. How much voltage and current is actually required to operate the solenoid?
The main reason for a clock not being suitable for batteries isn't the power required, it's the voltage. If that solenoid really does need 24V to actuate, then batteries are out of the question - a pack of sixteen AA cells would be ridiculous!
I think
@electrofix hasn't properly read and absorbed some of the posts in this thread - certainly not mine! Please, don't over think it. All you need is a solenoid, ratchet, and gearing between the minute and hour hands. All the rest is obsolete because its only of use when the clock is controlled remotely.
So, strip it down to the solenoid, and then test the solenoid to see what voltage it actually requires. This is far simpler than you seem to imagine. I've explained this already, but apparently I need to explain it again.
When you've done that, and found out how much voltage and current the solenoid needs, please report back and we can move on to the next stage.