The first thing you will need to do is go to the Swamp region.
You can do this by either walking the entire way from the Farming Village.

Or you can just fast-travel there with your Map.

Now you will have to walk the rest of the way.

The Antique Hat is located just inside the Jungle region. But WAIT!
As soon as you take the Antique Hat from the Monkey's fort...
The Monkeys will get angry at you and chase you while attacking you with ranged and melee weapons! So you'll want to RUN!