Prove that $p$ is the smallest prime that divides $(p-1)!+1$

Your solution is correct, and a slightly different approach (which ammounts to the same you wrote)

Hints:

(1) Wilson's Theorem

(2) For any prime $\;q\;,\;\;q<p\;$ , we have that $\;q\mid (p-1)!\implies q\nmid \left[(p-1!+1\right]\;$ ...