Actual experience with primitive peoples indicates they did not, and still do not, think this way, though.
Specifically in reference to archery, because you mention it:
George Catlin found that the best archers among Native Americans were considered to be better because they had more "sacred power". The concept of skill acquired through practice was, weirdly, unknown to them. One man was the better archer purely by virtue of having more "sacred power".
In the matter of fishing:
On the currently running TV show River Monsters, host Jeremy Wade finds that he is often required to consult a local shaman or religious figure of some kind before he starts catching the particular fish he's after. His skill as an angler means nothing to the local tribesmen: the way to catch a fish is to propitiate the river spirits or fish spirits.
So, while humans have always believed in cause and effect, the cause was more often than not considered to be ineffable "mojo" of one kind or another. If you cut down a tree to make a canoe, you have to have a ceremony first apologizing to the tree. If not, the canoe will sink or bring you bad luck somehow. Your abilities as a Naval Architect are moot.
It's doubtful cave men were doing much that we'd call science.