I'd much prefer to present the second-order Induction Axiom like this:
$$\forall X([X0 \land \forall n(Xn \to Xn')] \to \forall nXn)$$
with a genuine second-order quantifier ('for all sets' sounds too much like a sorted first-order quantification over sets). The corresponding first order schema is
$$([\varphi(0) \land \forall n(\varphi(n) \to \varphi(n'))] \to \forall n\varphi(n))$$
And the correspondence is plain. Given a second-order universal quantification $\forall X\Psi$ (where the initial quantification and its associated monadic variables are the only second order expressions in the sentence) the counterpart first-order schema is formed by deleting the initial quantifier and replacing every $X\tau$ for term $\tau$ by the schematic $\varphi(\tau)$.
If a second-order theory is such that any second-order axioms can regimented with single prenex second-order universal quantifiers (and associated monadic variables), then the theory will have a schematic first-order counterpart formed in this way by replacing the second order axioms with all the instances of the counterpart first-order schema. (In principle, there could be more complex cases, where we have a second-order theory whose second-order quantifiers run over many-place relations, not one-place properties and/or there are multiple quantifiers: then schematization would be more complex, but as far as I know [away from the library!] things go as you would expect.)
You mention that second-order arithmetic has a first-order counterpart where the second-order induction axiom is replaced by every instance of the schematic counterpart of the axiom. Another example is second-order set theory where a second-order replacement axiom has as its first-order counterpart the instances of the familiar first-order replacement schema, etc. For more, see e.g. Shapiro's Foundations without Foundationalism: A Case for Second-order Logic.