There is a Mongol campaign available in the Warlord Edition/Mongol Invasions version. You can select it from the full campaign section of the game. There are arrows (right and left) that allow you to select Sengoku Jidai, 1530, 1550, 1580, and Mongol Invasion. They are easily overlooked.

In the Mongol Invasion campaign you play the Hojo, and must defend all of Japan against the Mongols, who land in massive force on the island of Kyushu. Your troops will be spread all over Japan, so the challenge is getting them where you need them quickly, and building more as soon as you can.

Buddhist warrior monks, arquesbusiers, musketeers and other Japanese units are not available in this campaign like they would be in the others. This corresponds to the warriors that were present in this time period.

I've found that it's best to try and contain them on Kyushu. If you allow them to spread to the mainland, it can be very difficult to oust them. If you can prevent them from conquering a lot of provinces, they will not get as many reinforcements--the number of reinforcements they get is based on the amount of koku they pillage. Note that subsequent reinforcements WILL land on the mainland, so--as a good general should--cover your *ss ;).

Enjoy m8, I hope this helps.