This depends heavily on what your definition of "Noetherian" is. Are you working with the ascending chain condition on ideals?
If you start with an infinite nondecreasing chain $A_i\subseteq A_{i+1}$ of proper ideals (in a ring with identity), then $\bigcup_{i=1}^\infty A_i=(a)$ is a proper ideal containing all the $A_i$'s.
But $a\in A_j$ for some $j$, and that would mean that $(a)\subseteq A_j$. In that case, $A_j=\bigcup A_i=A_k$ for every $k\geq j$.
This says that all such chains stabilize, that is, the ring satisfies the ascending chain condition on ideals.
The same reasoning shows you that if you just suppose all ideals are finitely generated, then the ascending chain condition holds.
Are you using the maximum condition on ideals?
I think, perhaps, it was intended for you to use Zorn's Lemma to prove that ACC on ideals implies this. For if you are given a nonempty set of ideals of $R$, Zorn's lemma and the ACC together imply that every chain in the set is bounded, and hence the whole set has a maximal element.
The converse implication (maximal condition implies ACC) always holds, of course: given any chain, the maximal condition implies the chain has a greatest element, and that would make the chain stable.
Despite the name, Zorn's Lemma is just the Axiom of Choice in another guise. You are free to assume it holds, or not, and we often assume it. Using it is easy: if its hypotheses are satisfied, then its conclusion holds. To do this, you need to validate that ascending chains in a poset are bounded in that set, and then you magically get a maximal element somewhere in the poset.
Best Answer
By the Hilbert basis theorem, both $\Bbb Z[x]$ and $\Bbb Z[x,y]$ are Noetherian rings.
$\Bbb Z[\sqrt{n}]$ is just a quotient of the first ring, and since the quotient of a Noetherian ring is Noetherian, you have your answer.
$\Bbb Z[x/y]$ is the same as the localization of the second one at the multiplicative set $\{y^n\mid n\in 0,1,2,\ldots\}$. Since the localization of a commutative Noetherian ring is Noetherian, this ring is Noetherian.