One has an isomorphism
$$N_r: {\rm{Ext}}^r_Y(F, \pi^{\ast}G) \rightarrow {\rm{Ext}}^r_X(\pi_{\ast}F, G)$$
for finite etale $\pi:Y \rightarrow X$ between arbitrary schemes and any abelian etale sheaves $F$ on $Y$ and $G$ on $X$, without needing to appeal to an explicit procedure for an inverse map (though the argument below adapts to show that it is given by composing $\pi$-pullback with a canonical map $F \rightarrow \pi^{\ast}\pi_{\ast}F$ to which Mazur refers).

The idea is to reduce the problem to a sheaf-theoretic isomorphism, so one can then work locally on the base to pass to the case of a split covering that is easy. (This uses that we got away from the original setting in which $Y$ is required to be irreducible.)

Let's first recall how $N_r$ is defined (in the spirit of coherent duality). Since $\mathbf{R}{\rm{Hom}}_Y = \mathbf{R}\Gamma_Y \circ \mathbf{R}\mathscr{H}om_Y$ and $\Gamma_Y = \Gamma_X \circ \pi_*$ with $\pi_*$ exact (as $\pi$ is finite), we define $N_r$ via passage to $r$th homology on the effect of applying
$\mathbf{R}\Gamma_X$ to the natural sheaf map
$$\pi_{\ast}\mathbf{R}\mathscr{H}om_Y(F, \pi^{\ast}(\cdot)) \rightarrow
\mathbf{R}\mathscr{H}om_X(\pi_{\ast}F, \pi_{\ast}\pi^{\ast}(\cdot))
\rightarrow \mathbf{R}\mathscr{H}om_X(\pi_{\ast}F, \cdot)$$
using the sheaf-theoretic trace for finite etale maps (or even just quasi-finite etale separated maps) at the final step.

Thus, to show $N_r$ is an isomorphism for finite etale $\pi$ it suffices to show that the above displayed map is an isomorphism. But this task is of etale-local nature on $X$, and since $\pi$ is finite etale we can pass to an etale cover on $X$ to arrive at the case that $\pi$ is a split covering; i.e., $Y$ is a finite disjoint union of copies $X_1, \dots, X_n$ of $X$. Now $F$ amounts to a collection of sheaves $F_1, \dots, F_n$ on the $X_i$'s and the map in question is the natural composite map
$$\oplus_i \mathbf{R}\mathscr{H}om_X(F_i, G) \rightarrow
\mathbf{R}\mathscr{H}om_X(\oplus_i F_i, \oplus_i G)
\rightarrow \mathbf{R}\mathscr{H}om_X(\oplus_i F_i, G)$$
(using the addition map $\oplus_i G \rightarrow G$ at the final step) for any abelian etale sheaves $F_1, \dots, F_n, G$ on $X$. But this is obviously an isomorphism.