Assumption #1 No weapons, tools, or manufactured items other than the ceremonial dagger. No poisons, herbs, plants, minerals, or toxic soils. No traps, pits, or constructs. Picking up a branch on the ground and using it as a club or throwing a rock from the high ground would shame our warrior to the point of suicide.
To a degree, my ideas skirt that assumption of your rules. What's a tool? What's a weapon? If what you're looking for is some form of kung fu then, frankly, your question is wasted. You can research any martial art you want to rationalize overcoming the apes and explaining how to specifically attack an ape is story-based. Therefore, I'm also proceding on the presumption that there's a reason to attack apes without tools — because the fundamental nature of both evolution and intellect is to solve problems — and that inevitably means tools.
Worse, your civilization uses an ape's fang as a ceremonial dagger. Which means some overworked wife tired of hearing her warrior whine and bellyache about late meals grabs it to help prepare dinner. Suddenly the concept of kitchen knives is born — unless there's a cultural reason to never use that fang.
Assumption #2 There is a cultural reason to embrace Assumption #1.
Therefore, allow me to introduce some weaponless (from a certain point of view) solutions that are rationalized as some form of Tests of Bravery that utilize the circumstances of nature without handling any aspect of nature as a weapon.
The Test of Intellect
The indigenious peoples of the North American continent had an efficient way of providing supplies for large groups of people — they herded buffalo off a cliff. This was called a Buffalo Jump and sites have been identified in the U.S. and Canada. They used cliffs as tools to overcome a creature much larger and stronger than themselves.
Buffalo, being a herd animal, could be considered simple to slaughter in this manner. Get them running in the right direction and they'll pretty much bring the doom upon themselves. But your giant apes are a different matter. An individual fighter must either frighten the ape or bait it. The clever warrior must use the environment to distract the ape from realizing it's been led to the cliff edge. Thus, this is the test of the warrior's intellect.
The Test of Agility
If I can use a cliff as a tool, can I use a broken branch? This test doesn't test intellect as much as it tests agility. It could be called "The Dance." The warrior can easily lure an ape to a location where can be found a broken branch on a tree. The goal of the dance of combat is to cause the ape to lose its balance and fall onto the branch. While one could point to any number of Hollywood movies that show this kind of demise, if you really want to see something along the lines of what I mean, go watch Grand Sumo — but the superior warrior can do that without touching the ape!
The Test of Endurance
Everyone knows1 that the wild nature of the great apes means they really don't know how to hold their breath. Your warriors, on the other hand, train for months to embrace the risk to overcome the beast. Having found a suitable river or lake with exposed tree roots at the proper depth. Luring the ape into the water, your warrior holds the panicking and struggling ape under water. The struggle is brief, but intense, and the warrior must hold his or her breath throughout the ordeal.
The Test of A Fistfull of Dollars
Finally, the truly elite warriors combine intellect, agility, and endurance, to lure not one but two great apes into territorial conflict! The warrior must keep the two creatures engaged while not becoming a substantial interest to either. By egging on both apes, continually raising the stakes for both animals, eventually one or both are dead.
1 The Ministry of Meaningful Messages has been researching the veracity of this statement and can neither confirm nor deny that it is true.