I recall that when I was in high school that I read a text -which sounds more like a legend- about an Arabic Sultan/Caliph who on a trip trough his empire passed by an injured (wild) animal (don't recall what it was maybe a Lion?) and he decided to care about it in his Palace and later he established the first Zoo. I assume that the Capital was either Damascus or Baghdad, but I'm sure that that happened during either the Omayyad or maybe early Abbasid Dynasty!
History tells us that there definitely have been Zoos long before, but that is what I read in that text.
I would like to know whom that Sultan/Caliph was and whether this story has some back up and if so to what extend we might call this a Zoo?
I am afraid that this might only be a legend. From historical sources, we know that Zoos existed since ancient times as Menageries.
First such instance would be in Heirakonpolis, Egypt which existed roughly around 3500 BC. (Which is several thousand years before establishment of Umayyad Caliphate of Damascus or Abbasid Caliphate of Baghdad). In Levant, first such collection of exotic animals was created under patronage of King Ashur Bel Kala of Middle Assyrian Kingdom.
But Menageries were mostly reserved for Royal families and forbidden to public viewing. When we talk of the Zoos in modern sense of the word (Open to all), first such instance would be Zoos in Paris, Dublin and London.But you have already mentioned that you know there have been zoos before. The above context was to clear up that Abbasids or Umayyads did not establish "First Zoo" as per se.
Now moving on to specific Abbasid or Umayyad first zoos, the first description of a zoo is found in Hispanic Umayyads rather than Levant Umayyads. Madina-tu-Zahra, a Palace of Umayyads in Spain, had a zoo.. But in Levantine Umayyads, Caliph Abd-Al-Malk kept Lions at his court. Among the Abbasids, Al-Muktafi also loved to collect lions. Al-Amin Abbasid is known to have kept track of lions in his empire and had a dedicated force to protect civilians and livestock from them. There is no evidence for the said legend however AFAIK.