Designing a circuit to record timing of power cut
Use a microcontroller (For beginners, the arduino platform is very nice) and power it off a battery. Maybe an ATtiny is a good fit here, because it is small, simple and has low power consumption.

The device continuously monitors (directly via its ADC) the voltage of an AC-DC wall-wart (e.g. mains to 5V), and if it notices a drop to below some threshold, it saves a timestamp to its EEPROM (non-volatile memory). That's basically it. Reading out the device can be done in any way, e.g. Serial/USB. If you want precise timekeeping and don't want to sync the internal clock via the PC every few days, you'll also want an external realtime clock IC.

If you want it fancy you could make the wall-wart recharge the microcontroller's battery. But such simple circuits can usually work for a long time off a battery. You could also get a µC that can go into a very low power consumption sleep mode, and is wakened by a falling interrupt (not sure if that applies to the ATttiny). You might need to make a clear falling edge by throwing in a schmitt-trigger between wall-wart and interrupt pin.