Not sure what blog post you refer to but ...
You'd need a very super cap to do it. Then you'd have to control the cap voltage, not to drop it below 5V to ensure everything is stable.
When you add a shottky diode, you will drop 200mV+ from your cap rail.
The ATXRaspi continues to draw current from your cap, which will drain it.
FWIW back before there was MightyBoost and MightyHat and I think in the very early stages of ATXRaspi I put a lot of thought into this very idea. Eventually it never made it to production due to high costs and many complications and potential issues of using super caps in such a setup, with only very little ability to sustain a 1A load and for a very short time ... so it was NOT WORTH IT.
There have actually been some commercial products that backed up the Pi with supercaps, while cool and challenging in concept, too much to bother. A much simpler (and compact) scheme is a 5v boost with backup from lipo pack (IMO).