I found a pretty good explanation. http://www.nthuleen.com/102/hausaufgaben/explwortstell.html The bottom line - it just has to be accusatve if there's no preposition in front of the name of the day.
For action verbs, such as “to do”, the English simple present is used *only for
• the general present, if the action always or generally occurs in the given situation, as in “The sun rises on Sundays”. Because a unique time, “next Sunday”, is specified here, the general present cannot apply;
• the habitual present, if the action habitually or normally occurs in the given situation, as in “They go to church on Sundays”. Because a unique time, “next Sunday”, is specified here, the habitual present cannot apply;
• the narrative present, if the action is being described as one of a series of events in a story, as in “Solomon Grundy dies on Saturday, and he's buried on Sunday”. Because the listener is specified as the actor here, the narrative present can only apply in a situation where the speaker is coaching the listener. And because a future day, “next Sunday”, is specified, the narrative present can only apply in a multi-day story plot.
So you can say “What do you do next Sunday?”, but only in quite limited circumstances. For example, the speaker could be a coaching a politician about planned campaign tactics; or coaching an accomplice about planned moves in a long con job; or a coaching an actor in a partially improvised play with a multi-day story plot.