That is a lot of clipping.
You will want to use a loop in model builder or python to do things that way.
I would suggest, however, something simpler
like union-ing the two polygon feature classes
This will add the admin area attribute to the city area polys
now, using the attribute of the admin area,
and the city attribute,
you can select the cities in a particular admin zone
as a layer (or a selection)
and export or copy just those polygons to a new featureclass, (if you want)
Again, I would suggest just leaving all the intersected data in one FC
and making individual layers as you need them.
You say you have a cities "map" and an administrative boundary "map".
I assume you mean that you have polygon shapefiles or geodatabase feature classes:
the cities file has polygons with some sort of ID (name of the city, perhaps)
The admin area file has polygons with admin area IDs (the name of the country, for example).
In ArcToolbox, in the Analysis Tools > Overlay section is a tool called Union
read the help on that tool.
It will show you how you can get one set of polygons (in a new file)
that has the shapes and attributes from both the other two files.
Now you can use any of a variety of tools to select whatever polygons you want to use;
say, the cities in Utopia, using a Select By Attribute operation.
You can make a layer, using your query (example: cities in Utopia), that show just the features you want.
If you need separate files of each set, you can copy the data out to a new file
(but, as you write, you will then have more than 100 individual files).
The union is just one operation; you then get a data set you can look at different ways to get all the different 'maps' you want.
Doing it with clipping will require a lot more work:
first select or make the hundred or more layer files of individual countries
Then perform the hundred or more clips.
Best of luck