If I obtain a new character before their Celebration Missions begin, do I need to obtain them again after the missions start?

No, you do not.
The relevant mission will be considered completed, and you will be able to claim the rewards after the missions start.

However, please note that only missions requiring you to obtain the new character, such as “Obtain XX,” will be considered completed if the condition is met before the missions start. Other missions will not be considered completed, even if their conditions are met before the missions start.