Unfortunately the discharge characteristics of a capacitor are nonlinear so simple linear equations such as ohm's law do not apply.
See Capacitor Discharging for a reasonable explanation of the maths.
As Mike rightly says, the simplest solution would be to use a set of rechargeable cells to act as a back-up power source. The car and cells are ORed through isolation diodes to power the arduino, such that whichever presents the higher voltage supplies the power. When the ignition if ON, the car supplies the power, when the ignition is OFF the cells provide the power.