It actually depends on how you define Noetherian rings. The two most common definitions are:
A ring in which every non-empty set of ideals has a maximal element.
A ring which satisfies the ascending chain condition on ideals — every nondecreasing sequence of ideals is eventually constant.
The fact that (1) implies (2) is clear, but the implication from (2) to (1) is not provable without some weak form of the Axiom of Choice (specifically, the Axiom of Dependent Choice).
Now, using the first definition, it is trivial that every proper ideal $I$ in a Noetherian ring $R$ is contained in a maximal one. Take the collection $\mathcal{A}$ of proper ideals of $R$ that contain $I$. Note that $\mathcal{A}$ is nonempty since $I \in \mathcal{A}$. Therefore, $\mathcal{A}$ has a maximal element $M$, which must be a maximal ideal of $R$ by definition of $\mathcal{A}$.
Using the second definition, it is not provable without any form of choice that every proper ideal $I$ in a Noetherian ring $R$ is contained in a maximal one. It is tempting to proceed by contradiction. Assume there is no maximal ideal that contains $I$. Since $I$ is not maximal, we can find a proper ideal $I' \supsetneq I$. Since $I'$ is not maximal, we can find an ideal $I'' \supsetneq I'$. And so on, thereby obtaining a strictly ascending chain $$I \subsetneq I' \subsetneq I'' \subsetneq \cdots$$ However, each step of this construction requires choosing an ideal extending the previous one, infinitely many choices in total. In the general case, one needs the Axiom of Dependent Choice to justify this.
Note that the Axiom of Dependent Choice is only needed for the general case. For example, if the ring $R$ is countable then the process outlined above can be made effective. Let $x_0,x_1,\dots$ be a fixed enumeration of $R$. To pick $I'$, scan through the enumeration of $R$ until you find an element $x_i$ such that $x_i \notin I$ and $I + Rx_i \neq R$, then let $I' = I + Rx_i$ and continue in the same manner to find $I'', I''', \dots$ If $I$ is not contained in a maximal ideal, then such an $x_i$ can always be found and thus we contradict the ascending chain condition.