Thanks for both answers! They saved me a lot of searching and trying. I'll switch off the serial port now too while the power is down, that should put it down properly.
And for your questions:
septillion:
a) Why do you need two Arduino's for a simple alarm clock anyway?
Because my first version experienced too much interference over a 10m cable.
septillion:
b) Why on earth do you want to reset one of the two?
To synchronize after resetting the main one, which is the easiest way to prevent a clock overflow.