You'll hear about the Mongols around 1200-1210 I think and their invasion will start in 1220.
Like you I was extremely concerned about them, though far less directly threatened... I started to prepare pretty early, my plan was to block them on the mountains with some independent stacks. I had around 14 stacks. When they arrived at my borders though, they surpassed me in strenght (I guess, since I have manage to stay out of the conflict for now but who knows how long it'll stay that way). Since you're the first on their doorstep, I bet there is no way you can avoid them and you'll probably have to fight them on your own. Make sure you have several fortresses at your eastern borders, fortresses that will supply you in men. And this is not a loss of investment since you'll have to face the Timurids later on too. Prepare yourself with at least 15 stacks, taking in account the fact that they have excellent archer cavalry (better than the turkomans), great heavy cavalry (though inferior to your Royal Guard in terms of armour). Surprisingly they also have somewhat nice heavy infantry effective against armour. Your best options would be to fight them on favorable positions with mountains and hills or destroy them in siege by locking them on the gate with the boilinng oil. Careful though they often come with trebuchets.
On the other hand, there are some good news: they will be quickly in bankrupt since their troops are very expensive so they won't be able to recruit more soldiers. Try to destroy their armies one by one and avoid confronting their all horde together. Try to put a full guarnison in the the settlements that are close to the borders and surround them with 2 or 3 other stacks. And as you said, use the map, rivers with bridges and gap in mountains make excellent standing points. Don't underestimate them though. I suggest you save the game around 1210 or even 1200 and see if you manage them. Start building some army buildings (instead of farms and other in your fortresses) and train a maximum number of units.
Can't really add anything else though but good luck. I'm probably about to strike them myself with the help of my Kievan ally. And we would be grateful if you could post your reports and experiences about the way you fought the Mongol Invasion, since the best experience you can acquire is on the battlefield.
May Allah be with you in your fight!
PS: Why don't you try to request a Jihad against them? Should be great