Well this is an example of using nor without neither. But there are cases where it's perfectly acceptable to do that in English, as well as the reverse. However, this is not one of them. Not just a prescriptivist thing--I can't imagine anyone saying "I don't eat rice nor apples" in real life.
I love DuoLingo but many of the pronunciations, at least with Russian are terrible. Low-quality encoding and many words like ни here are chopped off on the end. As I've said in other forms I would happily pay a monthly subscription for good audio - especially since the site (and phone app) have so much awesomeness to offer.
I eat neither rice nor apples = я не ем ни риса, ни яблок
The Russian sentence as given may be idiomatic (I wouldn't know) but it seems ungrammatical. According to what we are being taught, the correct declension here would be the genitive-as-partitive. The partitive of рис is риса, while the partitive of яблоки is яаблок. All in my humble, underinformed opinion, of course.