Digital Clock - set time x hours ahead?
A little microcontroller can generate as many "button presses" as you want, causing the time to be set to whatever you want.  If you want to play with that, try using an arduino (because it's easy to get started) and connect an opto-isolator or reed-relay across each button on the clock.  By pulsing pins on the arduino, you can "press" the buttons on the clock.

However, "a little microcontroller" is exactly what is on that clock.  Why don't you modify the kit clock's software so that it comes on at 20:00 at reset instead of 00:00?  That would be much neater and require no additional hardware.

That chip (ATtiny2313) is an AVR, for which there are good free tools (compilers etc) available.  In fact, it is a smaller version of the chip (ATmega328P) present on the arduino.

You will probably discover than an arduino with RTC is a far more powerful, accurate and flexible clock than that kit, and cheaper.  It could trivially represent multiple timezones and show each timezone on a different display, for example.