Does the ADC-AC-LP1501-PSE come with a back-up power supply?

No. The LP1501 PoE Door Controller with Power Supply Kit (ADC-AC-LP1501-PSE) does not come with a back-up power supply. You will need to add your own back-up power supply. 

For recommended back-up power supplies, see What batteries are recommended for use with the LP1501 PoE Door Controller and Power Supply Kit (ADC-AC-LP1501-PSE)?.