On My 13th Birthday, I Became an Empress

Juusan Sai no Tanjoubi, Kougou ni Narimashita.
十三歳の誕生日、皇后になりました。

Source

Matsurika Kanriden
Matsurika Kanriden: Koukyuu Nyokan Kimagure Koutei ni Misomerare· 茉莉花官吏伝~後宮女官、気まぐれ皇帝に見初められ~
manga

Matsurika Kou is pretty content with her modest rise in position from a low rank palace maid to an official within the inner palace of Hakurou country. One day her boss asks her to take part in a practice marriage interview in place of a co-worker who got sick. During that fateful meeting, the wheels of fate will begin to turn...

Source

Juusan Sai no Tanjoubi, Kougou ni Narimashita.
十三歳の誕生日、皇后になりました。
novel

On her thirteenth birthday, Rian is sent to join the harem of the emperor of Sekisou Kingdom. However, the person receiving her is Akatsuki, who had usurped the throne through "legitimate means" to become the new emperor! Rian is made empress, as she is deemed "suitable," but even though they share a bed, Akatsuki treats her like a child, like a hot water bottle to warm his bed. To help Rian sleep, Akatsuki challenges her with a problem to solve every night!? (Source: Kadokawa, translated)

Parent Story

Matsurika Kanriden
茉莉花官吏伝
novel

Matsurika, a female officer in the Inner Palace, had a little special skill, and that was "a better memory" than others. One day, she accepted a request to become a practice partner for a marriage meeting of a certain son of a distinguished family. However, the one who appeared was the Emperor, Hakuyou, who wanted to try having a marriage meeting?! And interested in Matsurika's special skill, Hakuyou even said, "For now, go and pass the civil exam"...?! A Chinese-style Cinderella story between an emperor and a girl! (Source: Kadokawa, translated)