I'm sure that the system has a real time clock chip (maybe a DS1337) that only pulls a few microwatts when the rest of the phone is powered down. The alarm software sets the next alarm time on the real time clock chip. That chip will signal an interrupt line and will also signal the voltage regulator to power up the device. The phone on startup notices that the alarm fired and will notify the user. Any phone with the right hardware should be able to do this.