In some cases we have a word like of, which is av in Swedish. But for containers with content, more often we use med, which usually corresponds to with in English. So you could also say en kopp med kaffe. Av tends to be used more for things that are "constructed out of" something. Like, ett hjärta av sten = "a heart of stone". But, in general, we do use the of less. And of course it is often replaced by compound nouns as well.
No, there is a word for "of", but you don't use it when talking about a cup of coffee, a bowl of cereal etc. The word is "av" and you can read when to use it here: http://blogs.transparent.com/swedish/the-many-uses-of-the-preposition-av/