Average time to solve maze through backtracking
Your question on the exam is asking for average case analysis -- visiting all the cells to solve the maze is the worst case. 

Informally think this way: Whenever you move to a new cell, you can move to at maximum 3 directions, which one direction is to go back to your previous cell (this choice is discarded since it's of course not what you want to do to explorer the maze.) In the two other options left, one of the direction will lead you to the target, the other one does not. What's the expectation of find the target in each move? Does it hint something $O(n \log n)$?

Also try to picture this as you are exploring a tree structure. What's the average height of a tree?