Suppose we have an étale morphism between smooth quasi-projective (complex) varieties $X \rightarrow Y$ and assume that $Y$ is unirational. I am wondering whether we can somehow deduce that $X$ is unirational, or rationally connected, or uniruled, or has negative Kodaira dimension.

And what if $Y$ is just uniruled but non-necessarily unirational?

In the case where $Y$ is unirational and projective, there exists no non-trivial étale cover of $Y$. In fact, the fundamental group of a complex, projective, smooth unirational variety is trivial, see

J. P. Serre: *On the fundamental group of a unirational variety*, J. Lond. Math. Soc. **34**, 481-484 (1959). ZBL0097.36301.

finite, e.g., proper. However, the OP only indicates that the morphism is etale, not etale and finite. $\endgroup$ – Jason Starr May 27 '19 at 14:17anyprojective variety $X$ of dimension $n$; a general projection to $Y=\mathbb{P^n}$ is finite. Just take out of $X$ the ramification locus, you get an étale map to $\mathbb{P}^n$. $\endgroup$ – abx May 27 '19 at 15:34