Possible to use a Double Layer Capacitor (SuperCap) instead of Electrolytic?
You certainly could use a supercap instead of a normal cap.  But there are issues that need to be addressed.  In your case, the cap you chose will not work.

Like any cap, you need to make sure that you will stay under the rated voltage of the cap. You are good in this case. 

Your power supply must be able to handle the inrush current.  When you first turn on the device, the cap will be discharged and must be charged.  This charging current could be quite high, and your power supply will have to handle it gracefully.  This doesn't mean that the power supply needs to be rated for dozens of amps, only that it can successfully charge the cap without shutting down or blowing up.

Not all super-caps are equal.  Some are designed for powering very low power devices for a long time, while others are designed for providing a huge pulse of power for a short time.  The cap you picked is the former, not the latter.  Where this matters is in the internal resistance of the cap-- which is 80 ohms in this case.  80 ohms might not matter when you are powering an RTC which only consumes 1 uA.  But 80 ohms is absolutely HUGE in this application.  The voltage drop from that at 200 mA is 16 volts!  So at 200 mA the voltage out that you'll get from this cap is almost zero.  You will need to pick a different cap that has a much lower internal resistance.