Good news: Hungarian postpositions mostly take nominative. Of course there are "composed postpositions" by which I mostly mean words that have their meaning on their own and therefore they take some ending (e.g. "közel valamihez", pretty much like "close to sth"). Aside from that ("aside from", another one... this is not uncommon in English either), I can only think of one case and one postposition - át + superessive (-n). A folyón át - Through the river.
I mean... if you have a lot of suffixes and postpositions, at least don't combine the two, let it be all flat. :D Rather than having to remember postpositions combined with cases
You know, that's what people ask about the most. :) Also, the part I like the most.
Anyways, there are things I could make more accurate now.
Phrases like "közel valamihez", while they may be prepositional phrases in English, are actually more like adverbial phrases, and you can move "közel" around quite freely.
Also, I have found a preposition (not too common in Hungarian, as you can see :D) plus another postposition that involves a nontrivial case ending.
"Mint" is a word that has many meanings, just like "as" in English, they overlap a bit - "mint orvos" can mean "as a doctor". There is a suffix with similar meaning though: -ként, so "orvosként" is pretty much the same (fun fact, this suffix only has one form regardless vowels of the word it applies to, asztalként, emberként, it's all the same).
And the mentioned postposition is "valamihez képest". It has a comparative sense but it's not easy to translate exactly, it's something like "compared to" or even "for" (the following meaning of it according to Cambridge dictionary: "used for comparing one thing with others of the same type").