Is the runtime complexity of sorting $O(n\log n)$ or $O(n\log^2 n)$?
Algorithms are usually analyzed using the random access machine model. In this model, arithmetic operations on machine words take time $O(1)$. A machine word contains $O(\log n)$ bits, where $n$ is some natural parameter, say the length of the input (in machine words!).

Given this definition, comparison-based sorting algorithms run in time $O(n\log n)$ as long as the arguments fit in a constant number of machine words.

Sorting algorithms probably take more than $n\log^2 n$ time on a Turing machine, since Turing machines have no random access. Turing machines are not so interesting, however, for analyzing the running time of algorithms, except for special circumstances such as proving results on the Turing machine model itself.

In your particular case, you can actually sort the array in $O(n)$ using counting sort. The $\Omega(n\log n)$ lower bound only holds for some models such as the comparison model.