The above answer is unfortunately wrong. It is not true that you need to "specify tags before start downloading of OSM data." in order to be able to access other keys.
If there are missing attributes/keys, you can use the "OSM Attribute Selector", tool to add them to your Feature Class table. This tool is also part of the ArcGIS Editor for OpenStreetMap toolbox.
OSM XML files downloaded and converted using the toolbox contain a special binary field used as a key/value storage similar to PostGreSQL HSTORE. This field contains full tagging info for a specific feature.
By using the OSM Attribute Selector, you can add the missing keys/attributes to your existing Feature Class table, as that tool will extract the required information from the key/value storage and create a physical field for it.