How Can we make $O(N)$ order statistic queries with Fibonacci heap in $O(N)$?
Well, you can build a Fibonacci heap in $O(N)$ time. Assuming you will get the proposed order statistic also in $O(N)$ time for the index set $\{1,2,3,\ldots,N\}$, then we can sort the stored set in linear time.

You probably know that (comparision based) sorting takes $\Omega(N \log N)$ time. Since you have no information about the universe of your elements you cannot use tricks like bucket sort or sorting on the word ram to get a linear time sorting algorithm. So you can't solve your problem in $O(N)$ time.