The studies cited thus far are all observational and only show an association. They do not explain the mechanism. You are (unconsciously and because you're being lead by the nose by good ol' bad journalism) jumping to a causative relationship when none has been established, especially when you ask the question, "Did cavemen all die of prostate cancer?"
Keep a few things in mind:
1) Not everyone who eats animal fat gets prostate cancer. Thus there is no 1:1 causative relationship in the same way that getting hit with a baseball bat in the head with a certain force will break your skull or being irradiated with a certain dose will kill you.
Lesson: Never mistake association/correlation for causation. Establishing which event causes another is not a trivial task in human health, and it's very easy to put the cart before the horse when just working with observations.
2) The people who were studied were eating, generally speaking, the standard american diet and thus there are all sorts of confounding factors. The paleo approach is, almost entirely, based on saying, "This is observed in modern man on a modern diet; what if we didn't eat a modern diet?" It's possible that to get prostate cancer, you need to eat animal fat AND X, where X is not typically present in a hunter gatherer society.
Lesson: Never forget confounding factors; simple relationships are usually not that simple in reality. (don't forget to apply this to your favorite paleo dogma either)
To quote the NY Times article:
Both the writers of the editorial and the authors of the new study pointed out that the relationship between fat intake and the promotion of prostate cancer was complex and in need of much further research. Dr. Giovannucci said, "The findings need to be confirmed in similar prospective studies in different populations, and more research is needed into how animal fats might promote prostate cancer."
3) They cite "alpha-linolenic acid from animals" as the fat causing the highest cancer risk, but earlier in the article they claim that omega-3 fatty acids are anti-cancer. This is somewhat bizarre since alpha-linolenic acid is an omega-3 fatty acid typically associated with plant sources, not animal sources.
In the course of searching for some clarification, I came across a study from 2006 which directly contradicted that observation: http://www.psa-rising.com/blog/2006/06/alpha-linolenic-acid-no-impact-on-prostate-cancer/
So here we have two large studies, one asserting one thing, another saying in essence, "No association."
Lesson: Verify the details.
NOTHING in human health is simple. Even the basic logic of avoiding neolithic foods can only reach the status of a heuristic, not a principle. Something like "eat animal fat -> get prostate cancer" just doesn't pass the nuance smell test.