I'm coming from a differential geometry background (though I'm pretty familiar with category theory) and trying to learn a bit about ringed spaces.
Let $(X,\mathscr{O}_X)$ be a locally ringed space, and let $\iota_x : \{x\} \to X$ denote the inclusion of a point $x \in X$. Then based on my experience with smooth manifolds, I expect that we should define the structure sheaf of $\{x\}$ to be the residue field $\kappa(x)$ in order to get a locally ringed space structure on this point. In this case, I assume that the adjunct of the quotient map
$$\iota_x^{-1} \mathscr{O}_X \to \iota_x^{-1} \mathscr{O}_X / \mathfrak{m}_x = \kappa(x), $$
namely,
$$\iota_x^\sharp: \mathscr{O}_X \to (\iota_x)_\ast \kappa(x)$$
should be the comorphism for this inclusion. If so, this would mean that
$$(\iota_x)^*\mathscr{O}_X = \kappa(x) \otimes_{\iota_x^{-1} \mathscr{O}_X} \iota_x^{-1} \mathscr{O}_X = \kappa(x).$$
Is my reasoning correct here?
Yes, of course; locally, a morphism $Y \rightarrow X$ gives a map of algebras $B \leftarrow A$, and for the $A$-module $M$ the pullback is defined to correspond to $B \otimes_A M$.
In your case, it is $$ k \otimes_A A = (A/m) \mathop{\otimes}_A A = A/m. $$