Mobility is the winning edge in this game, and swords/cats have less than horsemen. That said, swords should work just fine if you take advantage of terrain and ranged. Because they get defensive bonuses, the swords ----> rifle path can do SOME things horse units can't do, they are just less effective on average.
Your key priorities are scouting and positioning. Even swords can set up flank beatdowns on flatlands if the AI has no follow-up. Archers can chew the AI units up from behind cover of the swords - a nice feature for something that is comparatively inexpensive and does not eat up resources. Just be careful, and use great generals, good promos, and possibly discipline adjacent unit bonuses to push up your strength multipliers.