Is a finite inverse limit of noetherian rings noetherian?

The answer is no.

Let $\varphi : k[x,y] \to k[x,y], f \mapsto f(x,0)$ and

$$A = \{ f \in k[x,y] ~|~ f(x,0) \in k \} = \varphi^{-1}(k).$$

$A$ is well known to be non-noetherian - $(y,xy,x^2y,x^3y, \dotsc)$ is not finitely generated - but it fits in the following cartesian square (the horizontal arrows are inclusions):

$$\require{AMScd} \begin{CD} A @>>> k[x,y]\\ @VV\varphi V @VV\varphi V \\ k @>>> k[x,y] \end{CD}$$