Connecting arduino to an existing circuit
Yes, This can be achieved without a transistor.

As the circuits would be on isolated power supplies: Arduino circuit on one and the clock on another, you can use a Solid State Relay.



Use the relay with your Arduino (5v or 3.3v) and wire the relay contacts parallel to the clock switch. This will isolate any power issues as clock will be powered with its own power and Arduino will be powered with its own.

You can program Arduino with proper delays to actuate the relay accordingly.

Hope this helps.