As an example: I'd like to know, when was the Key "tourism" added (or last edited) of this amenities:
http://overpass-turbo.eu/s/1N4
I tried with different APIs but didn't get very far.
Is there a way over any API or do I have to get as SQL Snapshot and somehow look for the last changeset?
You can use the history call of the main OSM API if you know the IDs of the objects wich you want to inspect. The resulting XML includes all versions of the respective object, thus allows one to determine when certain changes were made: