If $I$ is an ideal which is maximal among the ones that are not principal, then $I$ is prime.
This would mean that for all $f \in R$, $(f) \subset I$. Could I then use colon ideals? I was thinking maybe that for $P$ prime, $(P: (f) )= R$ if $f \in P$ or $(P: (f) )= P$ if $f \not\in P$.
Pretty much the same answer I gave to your other question (Maximal Ideal Must be Prime).
This time, you'll need to prove that $J = I + (f)$ is not principal if $J$ is not principal. The method of proof is pretty much the same.