You can also use Serre's theorem which says that kernel of the natural homomorphism from the mapping class group of $\Sigma$ to $\text{Sp}(2g;\mathbb{Z}/3\mathbb{Z})$ is torsion free, and therefore every finite subgroup injects to $\text{Sp}(2g;\mathbb{Z}/3\mathbb{Z})$. But that gives a [polynomial bound of degree $g^2$][1] compared to $84(g-1)$.


  [1]: http://groupprops.subwiki.org/wiki/Order_formulas_for_symplectic_groups