The line comes from a short exchange in a game. The character is surprised seeing the protagonist walk into a practice room that he's been using, and later says the following:
which i understand as an assertion that the room is supposed to be booked for him (for the rest of the day, perhaps? The みっちり is tripping me up a bit), but what is the meaning of "しよう" here and how does it connect to the rest of the sentence grammatically? Is it a する conjugation or a totally different word?