It’ a humourous way of putting “un petit moment à me maquiller”; it refers to a board game played by the French in the last century and probably still today; it is called “Le jeu de l’oie” (ref). In this game each of the squares corresponds to an activity and the speaker in her sentence identifies life to a series of activities well confined to compartments, as in the game.

“case” is “square”; “tour” is “turn”; as you throw dice in the game you get from one square to another.

The construction “case maquillage” is the common construction of compound nouns; it means “square where the make-up is applied”.

Un tour, especially un petit tour, implies a detour from your primary purpose/goal. Not a major detour, especially if a petit one, but not where you were originally meaning to go.

La case is a box, or more generally a topic, an area of interest.

So here what you have is something like ‘a small detour into the makeup area and I’m all spiffy, but very sleepy, to go meet my lover’.