Does the falling constellation refer to the sun constellation or the rising constellation?

The "descending constellation" is the opposite of the ascending constellation.

So as long as you know the rising constellation, you can know what the opposite falling constellation is.

The six relative constellations are shown below:

Aries → Libra,

Taurus → Scorpio,

Gemini → Sagittarius,

Cancer → Capricorn,

Lion-Aquarius,

Virgo → Pisces