Wait until 4PM. You may do anything you like until that time. The monarch will arrive at your tavern at 4, speak to them about your summons. They will make a request of you to find a way to immortallize them in the stars. Ponder this for a bit. Wait again until 9PM. At 9, head to the Judgement Zone. You'll notice an increase in the number of stargazers. Discuss with 2 of them.
Visit the forest after that and ask two of the stargazers there for inspiration. Visit the docks, then gather inspiration from the stars for at least one hour. This is done by selecting an empty bit of sand and selecting the special action. You'll feel so inspired that you MUST perform a song! Do so! Then gather inspiration from the stars for another hour.
Return to the tavern, and you'll have a choice to either write a poem or a play for the monarch. If you write a poem, you may do so and then read the poem to the monarch (you may have to travel to the castle to do so). If you write a play, you may do so and then invite the monarch (Via Messenger Post) to the Tavern. Once the monarch arrives, recruit them to be an actor and perform the play on the stage.