Finitely-generated algebra over Z

Take $A$ local (you already reduced to it), with $m$ the max. ideal. I claim that $A/m$ is a finite field. Suppose first that it has char. 0. Then we get injections $mathbb Z to mathbb Q to A/m$. By Zariski's lemma, $mathbb Q to A/m$ is finite, since it is of finite type.
Now (unfortunately I don't have it on me), Atiyah-Macdonald have a beautiful lemma which says that if $A subset B subset C$ are (comm.) rings, $A$ noetherian, $A subset C$ of finite type, $B subset C$ finite, then $A subset B$ is of finite type.

In our case, $mathbb Z to mathbb Q$ is of finite type, contradiction. Thus $mathbb Z/p to A/m$ is of finite type, hence finite for some prime number $p$. So $A/m$ is a finite field. Also $m^n = 0$ for some $n$ since $A$ is artin local. Finally, $m^i/m^{i+1}$ is a f.d. $A/m$-vector space (since $A$ is noetherian), so it is finite as well. And $|A| = sum |m^i/m^{i+1}|$.