If the Arduino runs fine until the motor turns on and its the motor causing the power interruption, and you want to keep the MCU supplied during the interruption, you'll want more backside then a capacitor.
I'd go with a LiFePo4 charger unit, a 6Ah LiFePo4 12V battery and connect a good quality 5V switching regulator to the battery to supply the 5V for circuit operation.
Or, go buy a UPS.