How to calculate areas of land cover from a File System Raster?

If you have the Spatial Analyst extension available, I think that the Tabulate Area tool will do what you're looking for. If you specify your polygon feature class/shapefile as the "Input feature zone data", your "zone field" as anything uniquely identifying the polygons, and your land cover raster as the "Input raster".

This will generate a table with the area of each category in the raster represented in the units of the raster. Note in my example below I had only selected two features in the counties dataset to reduce processing time, so it's worth remembering that having something selected in the "Input feature zone data" class will limit processing to just that subset.