You could also take the approach of using 5V thru 100 ohm resistor (limits current current from 5v source such as 7805 so rest of circuit can power up normally) to charge the caps in parallel, with a 0.3V diode in series.
Then run the caps into an LTC3525 boost regulator to create 5V for the processor only.
5V goes away, the diode keeps the cap from discharging to the rest of the board, the cap/regulator hold 5V up for the uC to do whatever it does to finish up.