It's possible to do this by brute force, using open affine covers for $X$ and $Y$:
$$ U_0 := \{ [x_0 : 1] \in X \}\cong{\rm Spec}\ \mathbb C[x_0], \ \ \ \ U_1 :=\{[1 : x_1] \in X \} \cong {\rm Spec} \ \mathbb C[x_1] $$
$$ V_0 := \{ [y_0 : 1] \in Y \}\cong{\rm Spec}\ \mathbb C[y_0], \ \ \ \ V_1 :=\{[1 : y_1] \in Y \} \cong {\rm Spec} \ \mathbb C[y_1] $$
On $U_0 \cap U_1$, we identify $x_0 \in \mathbb C[x_0]_{(x_0)}$ with $x^{-1}_1 \in \mathbb C[x_1]_{(x_1)}$. We make a similar identification between $y_0$ and $y_1^{-1}$ on $V_0 \cap V_1$.
Conveniently, we have $f^{-1}(V_0) = U_0$ and $f^{-1}(V_1) = U_1$. The morphism $f$ is associated with the ring homomorphisms:
$$ \mathbb C[y_0] \to \mathbb C[x_0] , \ \ \ y_0 \mapsto x_0^2$$
$$ \mathbb C[y_1] \to \mathbb C[x_1] , \ \ \ y_1 \mapsto x_1^2$$
The original structure sheaf $\mathcal O_X$ can be described as follows:
- On $U_0$: $(\mathcal O_X)|_{U_0}$ is the quasicoherent sheaf associated to the $\mathbb C[x_0]$-module $\mathbb C[x_0]$.
- On $U_1$: $(\mathcal O_X)|_{U_1}$ is the quasicoherent sheaf associated to the $\mathbb C[x_1]$-module $\mathbb C[x_1]$.
- On $U_0 \cap U_1$: the transition function is defined by identifying the element $x_0 \in \mathbb C[x_0]_{(x_0)}$ with the element $x^{-1}_1 \in \mathbb C[x_1]_{(x_1)}$.
So the pushforward $f_\star \mathcal O_X$ can be described like this:
- On $V_0$: $(f_\star \mathcal O_X)|_{V_0}$ is the quasicoherent sheaf associated with $\mathbb C[x_0]$, now viewed as a $\mathbb C[y_0]$-module, with $y_0$ viewed as $x_0^2$.
- On $V_1$: $(f_\star \mathcal O_X)|_{V_1}$ is the quasicoherent sheaf associated with $\mathbb C[x_1]$, now viewed as a $\mathbb C[y_1]$-module, with $y_1$ viewed as $x_1^2$.
- On $V_0 \cap V_1$: we identify the element $x_0 \in \mathbb C[x_0]_{(y_0)}$ with the element $x_1^{-1} \in \mathbb C[x_1]_{(y_1)}$.
Now observe that $\mathbb C[x_0]$ is a free $\mathbb C[y_0]$ module, by virtue of the $\mathbb C[y_0]$-module isomorphism $$ \mathbb C[x_0] \cong \mathbb C[y_0]. 1 \oplus \mathbb C[y_0]. x_0$$
So $(f_\star \mathcal O_X)|_{V_0}$ is a free sheaf of rank two. A similar statement is true on $V_1$. Thus $f_\star \mathcal O_X$ is a locally free sheaf on $Y$.
The sheaf morphism $i_\star \mathcal O_Y \to f_\star \mathcal O_X$ can described using module morphisms on the two affine patches. For example, on $V_0$, $i_\star$ is associated with the morphism of $\mathbb C[y_0]$-modules,
$$ \mathbb C[y_0] \to \mathbb C[x_0], \ \ \ \ \ y_0 \mapsto x_0^2,$$
which is injective, hence injective on all localisations at prime ideals. As the same is true on $V_1$, we see that $i_\star$ is injective on all stalks.
Finally, we describe the cokernel of $i_\star$. On $V_0$ this cokernel is the sheaf associated with the $\mathbb C[y_0].x_0$ component of $\mathbb C[x_0] \cong \mathbb C[y_0]. 1 \oplus \mathbb C[y_0]. x_0$. On $V_1$, it is the sheaf associated with the $\mathbb C[y_1] . x_1 $ component of $\mathbb C[x_1] \cong \mathbb C[y_1]. 1 \oplus \mathbb C[y_1]. x_1$. Notice that $\mathbb C[y_0].x_0$ is a rank-one free module over $\mathbb C[y_0]$, and $\mathbb C[y_1].x_1$ is a rank-one free module over $\mathbb C[y_1]$. So the cokernel of $i_\star$ is locally free of rank one. It only remains to find the transition function. On the overlap $V_0 \cap V_1$, we identify $1. x_0 \in \mathbb C[y_0]_{(y_0)}.x_0$ with $y_1^{-1} . x_1 \in \mathbb C[y_1]_{(y_1)} . x_1$. The identification $1 \leftrightarrow y_1^{-1}$ is precisely the transition function for the invertible sheaf $\mathcal O_Y(-1)$.
