I'm playing against Syzygy and didn't manager to complete 2 mysteries before getting 3 eldritch tokens on the red omen space.
After flipping the card and starting on the final mystery I completely ignored the previous unfinished mystery and also removed the eldritch tokens I had placed on the board.
I then realised that the Special Encounters on the final mystery would benefit greatly from having completed this mystery.
Am I allowed to continue on the Mystery from before flipping the Ancient One sheet? Or is this and all other mysteries obselete when starting the Final Mystery?
With other Ancient Ones, the normal Mysteries continue on and usually still need to be completed in order to claim victory. For example, Cthulhu won't appear on the board (so can't be defeated) until after you've completed the three Mysteries you needed to complete anyway, and Shub-Niggurath will appear but is immune to damage until three Mysteries are solved. So base rules (a) do not remove the existing mystery and (b) do not prevent further mysteries appearing (although thinking about it, maybe they should for reaching the normal official number).
Syzygy is obviously different since it flips as part of the standard progression of the game rather than as a directly bad effect of waking up. I can't recall the details, but the key rule is always "do what's said, don't do what isn't". Therefore if nothing on the sheet on either side says that the base Mysteries are cancelled when Syzygy flips, the mysteries will stay.
That said, I seem to recall the "on flip" text for Syzygy is something like "put tokens on this sheet equal to the number of mysteries solved", and the way to win is to get a certain total number of tokens on the sheet. Solving further mysteries won't put tokens on the sheet since that only triggers on flipping, so it's only if text on the Special Encounters explicitly makes reference to the number of solved mysteries that continuing with them would be useful in this case.