Add the layer to a dataframe that is set to geographic coordinates (perhap GCS WGS84), then you should be able to select that one. Currently, your dataframe and data are in the same coordinate system.
The same principle applies to http://desktop.arcgis.com/en/arcmap/latest/tools/data-management-toolbox/add-xy-coordinates.htm but you actually have to go into the Environments tab and set an output coordinate system.
In any case, you won't get all the coordinates for your polygon, unless you convert the polygon to points and repeat as above.
A simple calculate geometry in the table will also get you the x and y values, if the file is in a geographic data frame.
Many options, you might want to elaborate if the above doesn't cover your situations
