What you have about Ki-Rin in Volo's Guide to Monsters and Celestials in the Monster Manual is all you've got. Neither indicates that Ki-Rin are immortal or give any indication of their life-cycle.
Ki-Rin have appeared in every edition of Dungeons & Dragons except 4th and B/X starting with the Eldritch Wizardry supplement in 1976. This says nothing about their ecology.
The Advanced Dungeons & Dragons Monster Manual says:
The ki-rin are a race of aerial creatures whose hooves rarely touch the earth, for they dwell amongst the clouds and behind the winds. Females are never encountered, and ki-rin are always solitary. ...
Advanced Dungeons & Dragons 2nd Edition gives us:
Habitat/Society: The ki-rin are a race of aerial creatures that rarely set hoof on solid ground. Only the males ever approach the ground. No encounter with a female ki-rin has ever been recorded, although it is certain such beings exist. Likewise no young ki-rin has ever been encountered, thus details of their reproduction are unknown. Ki-rin are reticent about these topics.
Ki-rin sustain themselves by creating their own food and drink. ...
Dungeon & Dragons 3rd Edition Oriental Adventures says nothing about their ecology. The celestial template in this edition doesn’t make things immortal.
I did a quick search of Dragon magazine but the Ki-Rin is only mentioned in passing and doesn’t have an “ecology of” article.
So, it sort of looks like they have a "normal" life cycle but we still have no idea how long they live.
Ki-Rin are not a Dungeons & Dragons original, although the hyphen in their name came from D&D.
Kirin is, in fact, a very crisp, clean Japanese beer and the company that makes it.
However, before it was a beer it was a mythical creature called the Kirin in Japan and the Qilin in China, said to appear with the imminent arrival or passing of a sage or illustrious ruler. The earliest references to them date from the 5th century BCE, so if that guy is still going he's at least 2,500 years old. However, what I have been able to find (in English in non-scholarly sources) doesn't deal with their ecology - as mythological creatures, they don't really need one.
They became associated with the giraffe during the Ming dynasty when some of these creatures were traded to China from Africa. Giraffes can live up to 38 years which is a long time for a ruminant animal but a short time for powerful celestial spellcasters who live on the winds and herald the birth and deaths of luminaries but there is at least some real-life basis for using it.
Because kirin are often depicted with a single horn and hoofs, Europeans conflated them with the unicorn. I'm not going down that rabbit hole.