On the last day, you will get two letters: one from Clara, and another from Artemy asking for help. Choose either one and go and have a talk with Clara/Artemy. They with tell you that their respective Bounds are sick, and you must cure them.
Once all of the Bounds are healed, you will get another letter from Clara/Artemy asking to see you. Go to see them and tell them that you healed all of their Bounds. After that keep playing.
Once the clock hits 19:00 go to the cathedral and talk with Block. Tell him to listen what Clara/Artemy has to say. Wait until the day ends, and once the cutscene starts, you will get the achievement.