Is it possible to export a feature class field alias to a shapefile in ArcMap or ArcGIS Pro?

Answer

No, it is not possible to export a feature class field alias to shapefiles as aliases are not supported in a shapefile. A field alias can be added to a layer in the map which references a shapefile but is not saved in the shapefile data source. Hence, the field alias is lost when exporting a feature class to a shapefile.

However, a feature class field alias can be stored in an ArcMap map document (.mxd) or an ArcGIS Pro map file (.mapx) where the changes made for the field alias are saved in the feature layer. This is a temporary alternative, where the modified field aliases are stored in the map document or map file, and do not apply to the shapefile. Refer to ArcMap: Saving a map or ArcGIS Pro: Save a map file for more information.