ORing for Arduino Mini Pro
According to the data sheet you Atmega328 can go down to 2V power supply at 1Mhz so you can use a coin cell and a Schottky diode as a backup source. All your timings will be eight times longer but I don't think this would be an issue. Just change the prescaller clock while Arduino is powered from the main source, here is a topic about this. 

You can find plenty of low power techniques sources just googling , here is one. 

I think that the answer is wrong, the oscillator itself is not working under 2.7V so this will not work. 

Still, if the average power consumption is under 10mA (beep and rest) you still can use the Schottky diode and a larger capacitor on the 3.3V rail. The data sheet shows 0.3V voltage drop at 10mA which is still in the accepted range for 8Mhz.