Electronic – arduino – Super capacitor directly on the power supply

The basic formula for a capacitor is Q = CV. If this is differentiated you get: -

\$\dfrac{dq}{dt} = C\dfrac{dv}{dt}\$ and this equals current.

So, for a given current and a given capacitance the voltage rises at a rate of I/C.

You appear to be saying that the circuit that generates the current is "fixed" and so the only option I can see is make the capacitance "several" times bigger in order to lengthen the charge time.

Putting a resistor in parallel will not get you what you want because the capacitor will not charge up to the full voltage.

A more complicated solution is to create a constant current sink (and this is not a simple resistor as per your shunt resistor idea). The sink would be in parallel with the capacitor and basically this diverts current away from the capacitor making the net current into the capacitor smaller and hence increase the charge time.

You are going at this backwards. Always, always, always start by defining your requirements. Only then can you move on to considering solutions.

In this case, you need to first build a prototype dog feeding station. Only then can you get a feel for how much power/energy you'll need. With a nod to your idea of using supercaps, you can use motors which run at 5 volts. With a mechanism in hand, you can measure how much current the station draws (this establishes power requirements), and how long the motors run . From this you can determine how much energy you need. Remember to specify a worst-case power outage duration - if you size the system for 2 days but the grid is down for a week, your pups will not be happy. You'll want to run the feeder while you are home, anyways, in order to monitor it for reliability. After all, you don't want to find out that there's a weakness in the system by coming home to hungry dogs.

Only now can you start looking at candidates for energy storage. I suspect (very strongly indeed) that student is correct, and supercaps are not the way to go - they simply don't store enough energy to run motors for any length of time.

I'd also cast a jaundiced eye on solar power for your backup. If you are absent when a storm blows through, and the sitters can't reach your place, how will you depend on the solar cells not taking damage during the blow?

Frankly, I'd be inclined to start out assuming a biggish battery backup charged from the power grid would be the default position for your needs.