How do you get to the Observatory on Dragon Quest IX?

Answer 1

That depends on where you are in the story.

You start off in the Observatory, then you descend to the mortal world to start your job as Guardian of Angel Falls.

--- SPOILERS FOLLOW ---

You will return to the Observatory several times as part of the story. You might be able to return there via the Starflight Express (from the blue shining trees) if you have reached Alltrades Abbey.

If you've already beaten the story, then the Observatory is gone.