You are correct that you will not use the mikveh on the night of Yom Kippur; in fact, it will be closed that night. You should immerse the following night, when you break your fast. Since you will already have finished counting the seven clean days, you do not have to wear white underwear or do additional bedikot on the day of Yom Kippur.

If possible, you should prepare prior to Yom Kippur, much as you would prepare on Friday for a Motzaei Shabbat immersion. That way, after the fast, you would only have minimal preparations left. Alternatively, you can leave most of the preparations for after you break the fast.

It is a good idea to contact the mikveh ahead of time to check what their hours will be that night.