Age of Empires II: Definitive Edition

You have just over an hour game time (on Standard) before the Mongol invasion begins; I'd assume it's less on harder difficulties but maybe not. For 2 of the castles you can treb them down easily from outside the walls - the middle city from across the river, and then the southern (Polish) one from outside. The Bulgarian and Lithuanian ones you will have to take down their gates to get close enough.

I don't particularly recommend going through the marshy land to the west; it gets you a half dozen or so Elite Teutonic Knights and half as many Crusader Knights - it's fine to do if you have the time (I did) but it doesn't make much of a difference one way or the other. Just run around grabbing the villages (a few monks helps to convert Boyers to your cause so you don't need to train as many), keep training Elite Kipchaks as you have the food to tribute, and once you hit imp train a half dozen trebs to take out the castles. Once you have all 4 as your allies, you can delete the trebs - they won't do anything against the Mongol invasion and just take up population space.

Mongols invade from the east with a 400+ strong army of Steppe Lancers, Keshiks, and Manguidi with siege rams.