I'm trying to show that if $R$ is a ring with unity and $u \in R$ is not invertible, then there exists a maximal left ideal $L$ such that $u \notin L$.
I tried the following: I considered the poset of the left ideals such that $u \notin I$ and applied Zorn's lemma to obtain $L$, it remains to show that $L$ is maximal. This happens if and only if $1 \in L+Ru$. But I don't know how to prove that.
The phrase "there exists a maximal left ideal such that $u\notin L$" is ambiguous. Interpreted literally, the ideal has to be a maximal ideal in the whole ring, and $u\notin L$. But this is false: for rings with nonzero Jacobson radical, there are nonzero elements that are in all maximal left ideals (and they cannot be invertible.)
That is why it seems likely that your actual problem is to find a left ideal $L$ maximal with respect to not containing u.
That is certainly possible to apply Zorn's lemma to, and the conclusion is that there is such an ideal, but after that there is nothing left to prove about maximality: that was given to you by Zorn's lemma.