It suffices to prove that when $A$ has a unique minimal ideal and is an essential extension of it then $A$ is finite. This will follow from the Artin-Rees property along with the Nullstellensatz.
Edit: in more detail and focussing on the 'middle step': let $x$ be a non-zero element of $A$. Let $I$ be an ideal maximal subject to not containing $x$ - chosen using Zorn's lemma. Then $A/I$ has a unique minimal ideal $J/I$ and this is an irreducible $A$-module so finite by a version of Hilbert's Nullstellensatz. Now use Artin-Rees to conclude that $A/I$ is finite.