Metaphyzikal: I am wondering if you need to complete mysteries within a certain number of Sols to gain the achievement award for them? And if you finish them outside that window, and do them again and complete them under that time, do you get the achievement award? Another question is do all mysteries have achievement awards or only some of them?
As near as I can recall, there is no time limit on any Mystery.
However, with a few of them, especially harder ones with dilemma questions, it is possible to go down a path that will effectively "end" the Mystery, no way to proceed further, with out really "finishing it. In effect failing the Mystery puzzle. There are only a couple like that though.
There are also a couple where if you don't react quickly enough, or don't have the resources / developed tech to react strongly, great harm can be done to colony / planet / resources. Perhaps even enough to put your colony into a death spiral if it is small, or knife edged balance when it gets into the Mystery.
Those are rare situations, though.
If the Mystery tracking log in the upper left of screen clearly indicates that the Mystery sequence of events is finished, you will get the achievement, regardless of timing, colony size, etc.
BTW, if you want to postpone (most) Mysteries until you have developed / researched more, just don't go over 100 colonists! Disallow births in the domes for long periods, etc, etc. Almost all Mysteries that are not basically "terrain features" from touch down (there are either 1 or 2 of those) are triggered after 100 colonists...even if you immediately drop below that again.
I believe that the later DLC Mysteries DO NOT have Achievements in GOG, but that all Mysteries DO have Achievements inside the game interface.