For a smooth projective curve $Y$ over an algebraically closed field $k$ of characteristic 0, it is known that there exists a one-to-one correspondence between finite \'{e}tale morphisms $f:X\to Y$ of degree $p$ (a prime), and $p$-torsion elements of $\mathrm{Pic}Y$.


Is this correspondence still true when one replaces $Y$ into a smooth quasi-projective curve? 



Etale coverings of non-projective curves
What you say is not correct: when $Y$ is projective, $p$-torsion elements of $\mathrm{Pic}(Y)$ correspond to *cyclic* étale covers of degree $p$. In general, the exact sequence (for the étale topology)
$$1\rightarrow \mu \_p\rightarrow \mathbb{G}\_m\xrightarrow{\ \times p\ }\mathbb{G}\_m \rightarrow 1$$
gives an exact sequence 
$$1\rightarrow A\_Y^\*/(A\_Y^\*)^p \rightarrow H^1(Y,\mu \_p)\rightarrow \mathrm{Pic}(Y)[p]\rightarrow 0$$where $A\_Y$ is the ring of regular functions on $Y$.
So an étale $\mathbb{Z}/p$-cover $f:X\rightarrow Y$ determines a $p$-torsion element of $\mathrm{Pic}(Y)$, but this element may be trivial without $f$ being trivial -- it means that $f$ is obtained by taking the $p$-th root of an invertible function.


