1) You probably meant $\pi: X\to Y$ and not $\pi:Y\to X$. That way, any singularity that appears on a scheme of finite type over a field can be mapped to a smooth variety in a finite way. (This claim is implicit in VA's and Torsten Ekedahl's comments above).

2) A fairly general criterion for a singularity to be rational is given in my paper 
[A characterization of rational singularities][1]. In particular it covers your case in any dimension.


  [1]: https://projecteuclid.org/journals/duke-mathematical-journal/volume-102/issue-2/A-characterization-of-rational-singularities/10.1215/S0012-7094-00-10221-9.short