If I finish my third mystery during an encounter phase, but then Doom track reaches 0 during the related mythos phase, do I win the game? Or does the Ancient One awaken and I need to solve the final mystery?
The mysteries I've encountered are only solved at the end of the mythos phase, so, if doom is advancing, it would awaken the ancient one before mythos ends and the mystery is completed. If all mysteries follow this format, then my interpretation is you must flip the ancient one and follow the instructions on the back despite completing the requirements save for waiting until the end of the mythos phase.
From the Doom section of the quick reference:
When Doom reaches the “0” space of the Doom track, the Ancient One awakens.
From the Ancient One section of the quick reference:
Once the Ancient One awakens, the investigators must complete the Final Mystery on the back of the Ancient One sheet in addition to the other Mysteries.