Timer to control 1.5V clock
I suggest a simple setup (similar to my temperature and humidity sensor but less complex).

You can run from 3 x AA batteries, and have an RTC clock to keep track of the time. You can use a MOSFET to turn on your analog clock when required.

The processor can be asleep most of the time, using the watchdog timer to wake up and check the time from the RTC. Don't use an Arduino "board" unless you are running from a wall-wart because the voltage regulator, USB interface, and power LED, will consume too much power.