In some sense, it is common to do feature selection before you fit the ARIMA model, or at the very least, it is natural (in my opinion).
The problem is that there seems to be little development in automatic feature selection techniques for statistical time series models that can use exogenous variables (like ARIMA). Thus, it is not clear as to how we can do feature selection. To make things worse, auto.arima doesn't do any feature selection on exogenous variables, it just uses AICc to find the most optimal order of your model (in a stepwise fashion in its default setting). If you include exogenous variables in your model, they will always be included in all models in the selection process.
Basically, one way to do variable selection would be to try all possible combinations of exogenous variables, use auto.arima to find the "best" orders based on AICc, record this model's AICc (recall that AICc penalizes models that have large amounts of fitted parameters that do not increase the model's likelihood by a justifiable amount), and then pick the absolute best model out of all combinations of exogenous variables. Kind of a pain, and possibly very time consuming.
I hope this helps.