SPI is faster interface. I2C version can work as well.
You'll need a way to detect power loss (say a comparator that can generate an interrupt for you), and some capacitance to hold up the uC and FRAM power while you write a few bytes to it.
The capacitance can feed a boost regulator to make 5V from the decaying 5V of the capacitor, such as the attached.
Add a couple schottky diodes to isolate the cap/boost chip voltage from the regular 5V supply so the regulator isn't drained by the unpowered-power supply.
LTC3525fb voltage_boost.pdf (541 KB)