Something only implicit in that thread pointed to is the fast drop down of voltage during a capacitor discharge. A cap is not a battery! So only about 1/3 of the cap's energy can be used (from 5 to - say - 3.75 V) for an common Arduino. Using V-types will improve the situation of course.
5 sec @80mA are 0.4As which needs 0.1F @ 4V, hence a cap of 0.3F will do!