It follows from a theorem of Schoen and Yau and the geometrization theorem that $Y^3\times S^1$ admits a metric of positive scalar curvature iff $Y$ does.
On p. 9 of the paper, they define $C_3'$ to be the class of closed 3-manifolds three dimensional manifolds which do not admit any non-zero degree map to a compact three dimensional manifold $M$ such that $\pi_2(M) = 0$ and $M$ contains a two-sided incompressible surface with genus $\geq 1$ (i.e. a Haken 3-manifold).
Let $C_4'$ be the class of closed 4-manifolds $M$ such that every codimension one homology class of $M$ can be represented, up to some non-zero integer, by a map from a manifold of class $C'_3$.
Then Theorem 2 implies that if $M^4$ admits a metric of positive scalar curvature, then $M$ is of class $C'_4$.
Now, consider the 4-manifold $Y^3\times S^1$ and the homology class $[Y^3\times \ast] \in H_3(Y\times S^1)$. It follows from the geometrization theorem and virtual Haken theorem that $Y$ admits a metric of positive scalar curvature iff every finite-sheeted cover $Y'\to Y$ has $Y'\in C_3'$.
Any $M\subset Y'\times S^1$ such that $[M]=[Y'\times \ast]\in H_3(Y'\times S^1)$ is of class $C_3'$. However, there is a non-zero degree map $M\to Y$, hence $Y'\in C_3'$. So we conclude that $Y\times S^1$ admits a psc metric iff $Y$ does.