Making a super efficient line following robot that runs on super capacitors for as long as possible
The capacitor's voltage drops as current is drawn from it. Assuming constant power draw, the voltage curve (green line) should look like this (simulated with LTspice):-

In this example a 2F capacitor is discharged from 13.5 V to 0 V at a constant power of 5 watts, reaching 3.5 V in 34 seconds. At this point the capacitor has delivered 93% of the theoretical energy available. If the power draw is lower then the time increases in inverse proportion, eg. at 2.5 W it should last 72 seconds.
You could run 12 V motors directly off the capacitor, but since the end voltage is 3.5 V to get constant speed you would have to continuously adjust the PWM ratio to produce the equivalent of 3.5 V, and the motors would run at ~3.5/12 = 29% of full speed.
Alternatively you could feed the capacitor into a buck-mode DC/DC converter set for 3.3 V output. A good converter should have efficiency of &gt;=90%, which may be more efficient than than reducing motor speed with PWM. At 3.5 V the input current for 5 W output power rises to ~1.5 A, so you would need a converter rated for &gt;1.5 A.
However depending on how efficiently your robot can roll, you may need much less than 5 watts. The Polulu 50:1 MP 6V should do ~230 rpm at 3.3 V, drawing ~0.17A and 0.55 W at maximum power. But no-load current is only ~0.04 A per motor, so the actual current draw for 2 motors could be somewhere between 0.08 A and 0.34 A.
At lower motor current the consumption of your IR sensor array could become significant, so it might be better to use higher power motors that drive the robot faster, putting more of the total power into traction (assuming the robot can steer reliably at the faster speed) and less into the control circuit.
If you take the regulator and power LED off an Arduino Nano and run it from the internal RC clock you can easily get the current draw below 1mA, which is insignificant compared to the other components.