In case that this is not an April Fool's Day joke, I'll try to come. Sunday seems good enough (though 12:00 is a bit early-birdy...)

As this is at lunchtime, I'd vote for something like Koko, Südtiroler Platz: The food is good, it offers more than one vegetarian dish, and it is not only in walking distance to just about all the inner++ city of Graz, but also reasonably close to a parking place should anybody arrive by car.