2024 guide
Other solution no longer works - The game no longer gives you a recruitable catapult or 4 archers which are required to take down the castle quick enough.
When the match starts grab all your units, including the villagers and move them East to the red encampment. Run to the other end of the encampment, keeping your distance from King Takayutpi, and proceed to kill the elephants and other units there. Then have your villagers start building a wall in the camp and charge your army at Takayutpi to kill him before he even gets out of the camp. The walls will slow him a bit but aren't strictly necessary if you have enough units in his way.
If he slips past you you won't be able to take the castle out or kill him in time, and will need to restart.
However, if done right you can do this in just 3 minutes, and much simpler and faster than the old method of attacking the castle.
My first attempt at this strategy succeeded in 3 minutes 11 seconds.