Is there any simpler way to have for each sequence element the amount of succeeding larger elements than to implement an AVL tree?
You can use any balanced binary search tree data structure (doesn't have to be AVL tree).  Some such data structures are easier to implement (maybe a AA tree, left-leaning red-black tree, or splay tree?).  If you don't care about worst-case running time, it doesn't need to be balanced.  One pragmatic solution might be to use skip lists.

Are you sure the exam required students to implement their solution?  In my experience, exams in the algorithms and data structure courses I'm familiar with rarely require implementing the solution -- it is enough to describe the solution clearly.  A solution based on augmenting an AVL tree can be described in a fairly simple way: you say "take a AVL tree, and modify it in such-and-such a way", and that doesn't require re-describing all the details of how AVL trees work.  Only you know what the typical expectations are for the kind of exam you are looking at.