Let $k$ be a finite field, and let $G$ be the absolute Galois group of $k$, which is isomorphic to $\widehat{\mathbb{Z}}$. Let $\mathcal{C}$ be the category of $G$-modules. Then, we have the following:


For a finite $G$-module $N$, we have 
$$
Ext^r\_{\mathcal{C}}(N, \mathbb{Z}) \simeq H^{r-1}(G,~N^D),
$$
where $N^D:=\operatorname{Hom}\_\mathbb{Z}(N, \mathbb{Q}/\mathbb{Z})$ is the Pontrjagin dual of $N$, with the natural $G$-action by $(g\phi)(x):=g(\phi(g^{-1}(x)))=\phi(g^{-1}(x))$ for $g\in G$, $x \in N$ and $\phi \in N^D$.


Why is this true?



Galois cohomology of finite fields
As noted in R. van Dobben de Bruyn's answer, since $N$ is killed by some positive integer we can reformulate the problem as that of constructing natural isomorphisms ${\rm{Ext}}^i\_G(N, \mathbf{Q}/\mathbf{Z}) \simeq {\rm{H}}^i(G, N^D)$ for any $i \ge 0$ and any finite discrete $G$-module $N$ (with $G := {\rm{Gal}}(k\_s/k)$ for a finite field $k$). We will do this for any field $k$; finiteness is irrelevant.


It is convenient to work in the framework of abelian etale sheaves on ${\rm{Spec}}(k)$. From this viewpoint, a discrete $G$-module $M$ corresponds to an abelian etale sheaf $\mathscr{F}\_M$. The local-to-global Ext spectral sequence (which uses that sheaf-Ext into an injective abelian sheaf is flasque and hence acyclic for etale cohomology) is
$${\rm{H}}^i(G, \mathscr{E}xt^j\_k(\mathscr{F}\_N, \mathscr{F}\_M)) \Rightarrow {\rm{Ext}}^{i+j}\_k(\mathscr{F}\_N, \mathscr{F}\_M) = {\rm{Ext}}^{i+j}\_G(N,M)$$
for any discrete $G$-modules $N$ and $M$.


Now assume $N$ is finite, so $\mathscr{F}\_N$ becomes constant over an etale cover of ${\rm{Spec}}(k)$. It is then not hard to show by an erasable $\delta$-functor argument that $\mathscr{E}xt^{\bullet}\_k(\mathscr{F}\_N,\mathscr{F}\_M) = {\rm{Ext}}^{\bullet}\_{\rm{Ab}}(N,M)$ as discrete $G$-modules. Hence, the spectral sequence takes the form
$${\rm{H}}^i(G, {\rm{Ext}}^j\_{\rm{Ab}}(N,M)) \Rightarrow {\rm{Ext}}^{i+j}\_G(N,M)$$
for any discrete $G$-module $M$ and finite discrete $G$-module $N$.


Now set $M$ to be the constant $G$-module $\mathbf{Q}/\mathbf{Z}$. Then ${\rm{Hom}}\_{\rm{Ab}}(\cdot, M)$ is exact, so ${\rm{Ext}}^{\bullet}\_{\rm{Ab}}(\cdot,M)$ vanishes in positive degrees. The spectral sequence with this $M$ therefore degenerates to give isomorphisms
$${\rm{H}}^i(G, N^D) \simeq {\rm{Ext}}^i\_G(N, \mathbf{Q}/\mathbf{Z})$$
for all $i > 0$, as desired. (To be really useful one should address $\delta$-functoriality in $N$, but this was not requested in the question.)


