What they said! The distance might be a little long-ish on foot (as in, one or two miles walking in semi-urban, not very pretty zones? I think?), but you can definitely cross the Rhine, which is the border, on foot, and you will not need a passport. There are no checkpoint anymore (though I personally always carry ID when I change countries, no matter how easy it is to cross borders).

As for things to do, you will see if you look at a map of Strasbourg (using googlemaps, say) that the center of town is sort of an enclosed oval within rivers (a little like the islands on the Seine river in Paris).

This area – the center – is not too big at all, and it is full of nice places and squares and medieval streets, very nice to walk around in. The cathedral in particular is a must see; the area called ‘La Petite France’ is renowned for its charm as well.
The Museum of Modern Art is near that area, actually, and a good visit, both for the building and the collections (for example, go to the roof / terrasse café, the view is nice, and you can go there without maying the museum’s entrance, iirc).