Don't connect the solenoid to the Arduino directly as the Arduino can only supply 40mA at the very best of times (the recommended maximum is just 20mA).
It would be useful to know exactly what the solenoid needs in terms of voltage and current.
However, there are a few ways of doing this:
Use a separate power supply (or at least nothing connected to your Arduino board) for the solenoid
Switch the power to the solenoid using a relay module (and you will get a nice satisfying 'click' every minute just like a clock!)
Use a transistor (NPN or a MOSFET) to supply the solenoid with power. The base of the NPN transistor (or gate of the MOSFET) is connected via a small resistor to the output pin on your Arduino.
You will need to know how long to switch the solenoid on for (eg 100ms, 1 second) so it does its thing to the clock.
Are you running the Arduino on a mains adapter, USB or battery?
Finally, that 'blink' sketch is not the most accurate in the whole world; once you have it all up and running you will need to fine tune the delay as some time is taken up by the program code itself - not normally a problem but for a clock it would be a problem as it would run s-l-o-w!
Well, there's a few pointers, you'll have to come up with what you really want to achieve here before anyone can really help you further, I suspect.
BTW if you want to look at a relay option (just for background watching) I've just posted a video all about relay modules, it's #18 in the web address given in my signature below. Yes, another shameless plug!
Use a logic level power MOSFET. Add a resistor in series with the clock to limit the current to about 200mA. I suggest using a 1W resistor even if the pulse is short.
Add a kickback diode across the clock to prevent damage due to inductive spikes.
Instead of a MOSFET you could use a bipolar transistor with about 220Ω resistor in series with its base.
I recently built a power supply with standby battery for a Shortt-Synchronome clock (precision regulator unit only).
Sivoodoo:
The solenoid on the clock operates at about 1 volt and 200mA.
That implies a coil resistance of five Ohms.
It all sounds very odd to us. Common 1.5 V battery clocks would operate about that voltage, but surely not that current; more like a few milliamps for a few milliseconds. Some appear to alternate with an H-bridge.
True "slave" clocks would operate at a higher voltage. You need to describe more clearly what this clock is - provide a perfectly focussed picture taken in daylight.
Paul__B:
That implies a coil resistance of five Ohms.
It all sounds very odd to us. Common 1.5 V battery clocks would operate about that voltage, but surely not that current; more like a few milliamps for a few milliseconds.
I measured 10Ω on a Shortt-Synchronome clock from the 1930's but that probably included two electromagnets: for the dial and for the pendulum push mechanism. That clock needs more like 4V at 400mA but only for a fraction of a second every 30 seconds. I built a constant current supply to test it but it's now working via an adjustable voltage regulator.
Archibald:
I measured 10Ω on a Shortt-Synchronome clock from the 1930's but that probably included two electromagnets: for the dial and for the pendulum push mechanism.
The clock originally used cells in a shed down the garden. I never saw them and do not know their chemistry but I'm told they needed weekly maintenance. I guess they were Leclanché cells in glass jars.
I connected the solenoid to my power supply and slowly increased the voltage until the solenoid operated. Just over a volt and the amp meter showed about 200mA.