As a non-Melburnian too, here's what I'd recommend:
There's plenty of options on and near Brunswick St (Veggie Bar, Yongs Green Food, Trippy Taco). The Organic Food and Wine Deli opposite the Flinders St train station is also pretty great.
Also try the boreks or wraps from inside the Victoria St Markets. You'll no doubt bump into one of the Animals Australia crew there as it's a fave place to grab lunch
And non-food related, the Jewish Holocaust Centre is really worthwhile.