Pressing a physical button when a signal is given

I need to be able to press a physical button when something receives power. I have a "LED USB clock fan" that turns on and off by pressing a button, which I am using in my project, and I want it to turn on automatically at certain times of day. I have the fan plugged into a timer that only lets power through at certain times of day. However, I still need to press the button while the power is on for the fan to turn on. I was wondering if there was a good way to press the button on the fan (probably using a motor) when the power turns on, or if I just need to take the fan apart. I would prefer not to take it apart, however, in case I want to use the fan in other projects later. The fan is stationary, I just need something to apply enough force to press the button. The button takes I can do any programming necessary, I just can't think of a way to press it. Any Ideas?

Thanks!

Let the fan be connected via a relay between the timer and the fan. Then an Arduino can activate the relay.

A servo with small arm attached can do the button pressing for you.