On Tisha B'Av itself, bathing is prohibited, even if not for pleasure. You can either prepare prior to the fast, or delay preparation until after nightfall.

In general, when one cannot prepare during the day on a particular occasion (such as Shabbat or Yom Tov), there is debate among authorities as to whether it is better to prepare the previous day or the night of immersion. There is no clear cut answer. The issue is when you will be able to do the best unhurried job. In your case, it sounds like you should do as much of the preparation as possible on Monday, taking a quick shower and combing through your hair at the mikveh on Tuesday night.

Please note that preparation for the mikveh is not viewed as bathing for pleasure and therefore the usual preparation routine is permitted during the nine days. The restrictions of the nine days, including the prohibition of regular bathing, apply both Monday afternoon and Tuesday night.

Often, the mikveh will remain open later the night after Tisha B'Av to allow women to eat before going to the mikveh. Just be sure to brush your teeth well after eating. If there is not time to eat at home, then pack something to take with you.

For additional discussion, please see our posted question Preparing for mikveh after Tisha B'Av.