Is the Katětov extension of $\Bbb N$ zero-dimensional?

Your argument is correct, but the space you describe is not the Stone-Čech compactification! Indeed, you can see quite quickly that it is not compact: for each free ultrafilter $p$, the set $\{p\}\cup\mathbb{N}$ is open, and these form an open cover with no finite subcover.

For the Stone-Čech compactification, the basic open sets are those of the form $U_A=A\cup\{p:A\in p\}$ for $A\subseteq\mathbb{N}$. (Or, identifying points of $\mathbb{N}$ with the principal ultrafilters, you just take the set of all ultrafilters that contain $A$.) It is immediate that these are clopen, since the complement of $U_A$ is just $U_{\mathbb{N}\setminus A}$. (Of course, it is not immediate that this space really is the Stone-Čech compactification of $\mathbb{N}$, but that's a longer story.)