It's the button to deactivate the alarm. It's older technology, where pushing the button in physically stopped the ringing of the mechanical device.

"Pushed in the stop" is a bit of a strange phrase (in US English anyways), but I believe the author is saying that Robert turned the alarm off - or he pushed the "off button" or "stop button" on the alarm.

Apretó el botón que apaga la alarma.

That would be the stop button on his alarm clock.

I would think it means pushed the "stop alarm" button on his electric or maybe mechanical alarm clock so as to turn off the alarm. Definitely not a cell phone alarm clock.