Maybe I will just answer the specific question the OP is interested in by giving a reference of sorts (in case it is helpful to the OP).
In Bourbaki Éléments de mathématique: groupes et algèbres de Lie. Chapitre 9 Exercise 3 one is asked to prove that $\mathrm{Out}(PSU(n))\cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 3$.
It seems to me that in this case one can show this explicitly. The non-trivial non-inner automorphism is represented by the Cartan involution $A\mapsto ((A)^{-1})^T$ on $PSL(n,\mathbb{C})$ which simplifies to $A\mapsto \overline{A}$ on $PSU(n)$.