I think the problem is that the AI tends to fight wars in a more defensive style. Since there is no longer any uber-defensive unit, it makes it difficult for the AI to just turtle in their cities. Also, the addition of bombarding ability to nearly every unit means I don't have to worry about going for siege weapons at all. They're handy in the late game (although maybe even overpowered), but not necessary even then.
Given that statement, I would identify two other problems:
1. The buildings aren't numerous enough or powerful enough for me, the player, to worry about whether to build them or not. Either I don't build it because it won't do me very much good, or I build it quickly and then have nothing to build for a while again. This may be something unique to Armenia - after all, they seem to have tons of production with all those hills. Still, the AI is building all those buildings that don't do them much good (and may hurt them, like the walls and castle), while I'm sitting around building units. And keep in mind I normally err on the side of building things.
2. I felt like the AI didn't have enough workers, but that could be a vague mistake.
As for units: in my first war, which netted me Alp's capital and Jerusalem, I used spearmen. Once they got a few other units killed off and had the city attack promotion, they were unstoppable with their ability to bombard defenses.
For my second war, I did use nakharar, although I was still relying on spearmen then also, since I needed them for fighting the other spearmen.
Third war (with Batu): a mix, but for most of the actual combat I used foot knights, macemen, and pikemen. As far as I could tell, Batu didn't have a single maceman around, so my foot knights with city attack promotions could just tear through his defenders. Any knights he sent around outside the cities were easily picked off by my pikemen.
So what would I recommend? Maybe: make a few more buildings or make the existing ones a bit more powerful. Also possible: remove the bombard ability from a variety of early units. I wouldn't have been able to take those early cities with spearmen if they couldn't bombard (I'm not making that up, am I?)
however, I'm still having fun, and I'm a bit worried about how I'm actually going to get to any of the victory conditions at this point - it's not so easy, after all, that a domination victory is just around the corner. I am just about at the end of the tech tree, though...