On second thought, the translator's note about the book likely being a study guide for high school entrance exams might be inaccurate. I forgot that Yuuri's elite private school is
explicitly mentioned in chapter 6 of the serialization to be an elementary school, Adele Elementary. I suspect he's in grade 6, which would make him around 12.
It also could be that the school is so advanced that the students have the option of skipping junior high entirely (not sure if that's realistic for Japan, then again, the country in which the manga even takes place is ambiguous) so that the translator's note is correct and Yuuri is trying to get into senior high as a child prodigy. Or, perhaps, the junior high school that Yuuri wants to get into, possibly on scholarship, has an entrance exam of its own.