What would constitute being "born under this sign"? Probably not Sun-sign Astrology, although I suppose it's possible.
The Kali Yuga, by most accounts, is ending, not beginning. It's about a separation between the Mundane, materialistic perspective and the Spiritual and has resulted in Spiritual darkness for over 5000 years. So, prophecies of an uplifting Aquarian Age are in line with the transition OUT of the Kali Yuga which is already in progress.
It means Aquarius rising. As can be seen in the rest of the chapter, the qualities depend on the condition of the ruler. These seem to be on average, due to the malefic nature of Saturn. However, just as with Aries Rising, if Saturn is in his sign and bound, angular and with benefics in aspect, one will not at all possess most of these qualities except crafts, popularity and generosity for example.
While some Hindus might think that the Kali Yuga is ending, this is not the case with Christians and Buddhists.
The sign of the Sun has almost no effect in traditional astrology by itself. One of the signs of ''uplifting and progress'' of modern times.
Christians are waiting for the Second Coming, as prophesied in the New Testament. Many Hindus are of the belief the Kali Yuga is ending, and some believe it has ended already, and that we're now in a "transition period".
Both Aquarius and Cap value structures and systems. Cap wants materialism, reputation, and wealth. Aq wants structures that are impersonal and without a profit motive. Both are into authority, but Cap authority is based on status, whose judgments are rooted in efficiency and function. Aq on the other hand judges structures based on intellect, principles and ideas. So Cap is kind of like the judge and efficiency monitor of the material world whereas Aq does it more on the conceptual plane. Caps change their judgments when better efficiency models are introduced, whereas Aq change their judgments when things fit into their mental systems.
Aq affects human systems in a more broad, diffuse way... an 11th house way. Networks, collectives, information connections. Cap is more concerned with material exchanges. A corporate exec is a good symbol for Cap, whereas the dean of a university would be a good Aq one.
In esoteric astrology, the higher purpose of Cap is to transcend personal gain and begin to transmit the Divine purpose into material systems, through their top-level access to human hierarchies. That is how Cap's Saturn remakes the world. Aq on the other hand has to transcend its mental fixations about "correct knowledge" so that human knowledge can be fairly accessed by everyone transparently. The internet is an example of an Aquarian Age technology... it has democratized all knowledge. Nobody holds all the books.
I honestly think that Saturn rules Aquarius because astrologers were just like: "Well we 12 signs, and only 7 planets...uhh...let's just give Saturn to Aquarius because it's just as cold and detached as Capricorn.
Thankfully, the astrologers who devised the system thought deeper than that.