Translation:Zij is de zus van mijn vader.
I wrote "Zij is de zuster van mijn vader". I know it's not used often anymore, but is it actually wrong?
It's not wrong, zuster is just getting out of use and you shouldn't be afraid it will be wrong anytime soon. E.g. a word like schavuit which is way further out of use, is perceived as odd when somebody uses it, but at the same time it isn't wrong to use it, people just hardly ever do that. :)
Why is van used over op? Are they used in different contextes or are their meanings fundamentally different?
'van' and 'op' are two different prepositions in dutch. 'Van' would be translated as 'of' (de moeder van mijn buurvrouw - the mother of my neighbour). The word 'op', could be translated as 'on' or 'at' (there are more definitions and exceptions but just to give you an idea). For example: Op maandag gaan wij altijd naar de kerk - On monday we always go to church, Ik ben op school - I'm at school. In this sentence 'of' is used in the same context as the example i gave and 'van' is therefor the only suitable preposition to use. I hope this answered your question a bit.