В эту минуту я нахожусь с тобой.
В этой минуте, которую я проведу с тобой, мое утешение.

Russians diffirintiate easily because we always can ask "Когда?" or "В чем?" and it helps us to understand what case is used. But it won't help foreigners to understand the difference. I tried to look at this question through the rules when cases are to be used.

Acc

In certain expressions describing duration or frequency of an action,
Prep

To indicate the time of an action with wordsгод (year), месяц (month), неделя (week)
В эту минуту я нахожусь с тобой - duration?
В этой минуте - моё утешение - time of an action?

It still seems confusing as for me they seem to mean vise versa.