As formulated, there are many counterexamples, although I suspect the OP has a slightly different question in mind. For instance, let $Y$ be $\mathbb{P}^2$, let $X$ be the blowing up of the diagonal in $Y\times Y$ with either of its natural morphisms to $Y$, and let $Y'$ be the nonreduced scheme from Exericse II.5.9 of Hartshorne's "Algebraic Geometry" (I hope that is the right number -- I only have a Russian edition in front of me).

If you assume that $Y$ is affine, then by Exercise II.8.6 (hopefully), there is a retraction, say $r:Y'\to Y$. Thus you can pull back the smooth, projective morphism $f$ by $r$.

As formulated, there are many counterexamples, although I suspect the OP has a slightly different question in mind. For instance, let $Y$ be $\mathbb{P}^2$, let $X$ be the blowing up of the diagonal in $Y\times Y$ with either of its natural morphisms to $Y$, and let $Y'$ be the nonreduced scheme from Exericse II.5.9 of Hartshorne's "Algebraic Geometry" (I hope that is the right number -- I only have a Russian edition in front of me).

However, if you assume that your scheme $Y$ is affine, then by Exercise II.8.6 (hopefully), there is a retraction, say $r:Y'\to Y$. Thus you can pull back the smooth, projective morphism $f$ by $r$.