$$n^2 = 3^m + 1182 - (n - s(n)). \qquad (1)$$
If $m \ge 2$, because $n - s(n)$ is divisible by 9, but $3 \mid 1182$ and $9 \nmid 1182$, from (1) follows that $3 \mid n^2$ and $9 \nmid n^2$, false. Similarly, if $m = 1$, we obtain that $n^2$ is divisible by 3 and not by 9, false. Therefore $m = 0$, and $n^2 + (n - s(n)) = 1183$. (2)
If $n \le 9$, then $s(n) = n$, whence $n^2 = 1183$, false. Consequently, we have $n \ge 10$. From (2) we deduce that $n \le 34$, therefore $n = \overline{ab}$, with $a \in \{1, 2, 3\}$. Studying all the situations, we find that the only solution is $m = 0, n = 34$.