If you don’t hava a lot of luggage with you, I recommend that you stroll down to ‘Kristian F Møller’ - it’s a bookstore with an ok selection of Danish and English books - and then walk a bit further down to Latinerkvarteret (the Latin quarter) where there are many cosy little places. La Cabra is a good place for (expensive but excellent) coffee. But you can find many nice little places in that area.

There’s also Løves bog- og vincafé - it’s a bit further. Salling - a local department store - also has a nice café called Salling Rooftop.

For food - if you want it not too expensive - I’d say you go to the Aarhus Street Food market, which is just downhill from the train station towards the bus station. If you’re looking for something more fancy try Juliette in Jægergårdsgade - or one of the Burger Boom places. Aarhus has some excellent restaurants, but not sure which ones are open for lunch. Maybe Mefisto?

Edit:

If you’re in the mood for ramen try Gaijin Ramen - I’ve also heard good things about Takumi but I haven’t tried it.