First, thanks to dutchfire for posting this game. It was a totally different look for starting a Civ game.
First game I wasn't sure what lay ahead, so I tried taking most of the cities I encountered, thinking I might need to produce something to get to the goal. Didn't turn out that way and I dawdled in around 1550 BC.
Second game I decided to try to drive straight to the goal. This was working well (too well, sorry to say) when I made a mistake - I captured a city. Then thinking I had no real use for it, I let someone take it from me and then I lost. So, you can complete this with zero cities, but if you take one and lose it, you lose the game.
What's needed? I would say more advanced enemy units blocking the path. Spears and catapults are not worth worrying about, especially since the AI doesn't seem to know that the only thing that counts is killing Khan. Why not give the AI Cavalry or something so we need to plan very carefully to avoid or distract them?