Example of a commutative noetherian ring with $1$ which is neither domain nor local and has a principal prime ideal of height $1.$

Let $k$ be a field and $A=k[x,y]/(xy)$. Then the ideal $(y-1)\subset A$ is a principal height $1$ prime. (Note that this ring is not local, and the non-localness is essential to the example in that if you localize at $(y-1)$ then the ring becomes a domain.)

As a hint to what's going on here that can't happen in the local case, notice that $(y-1)x=-x$, so $x$ is divisible by $y-1$ arbitrarily many times. In a local Noetherian ring this would imply $x=0$ by the Krull intersection theorem.

Let $R$ be a noetherian ring with principal prime ideal $P$ of height $1$.

Then $R\times R$ is Noetherian, non local, not a domain, and still has a principal prime ideal of height $1$: $P\times R$.