As it says in the HorseArcherReadme.txt file, "if you try to add a new unit to a saved game, civ3 sometimes throws a wobbly"!
However, if you really are desperate to see the unit in action, then you could rename all the Arabian Horseman .flcs to say, 'Horseman', then slip them into the Horseman folder. Then each time a Horseman is built you will see an Arab on horseback instead of the civ3 animation (don't forget to make copies for the original Horseman .flcs first of course!). This works because as far as civ3 is concerned, nothing has changed as all the names of all the files are the same both before and after the game was saved.
But creating new units, and sometimes even just changing the stats, in a saved game can cause problems.