Assume a temperate environment (forests, hills, rocks etc. with, say, spring like weather) that is neutral (neither one is more familiar with it than the other). Who wins?
Like an astronaut with a tribeam plasma rifle? A caveman would be stronger but he would be unarmed.
No weapons other than ones they can make or find.
Is this one of the astronauts from the 60s where each one was an athletic genius fighter pilot or the more recent ones who tend to just be smart?
Caveman is likely to be smaller, dumber, and in poorer physical shape due to a lifetime of questionable nutrition. Astronaut may or may not be trained in martial arts, but caveman won’t be.
If I had to put money one way or the other, I’d bet on the spaceman.
I’m pretty sure a caveman doesn’t know how to play backgammon, so my money is on the astronaut.
Simple hand to hand combat? The caveman wins. He’s tougher and ready to go for the jugular immediately, he’ll attack with an adrenalin rush that makes him like a wild animal. Make the situation any more complex and the astronaut is likely better prepared to strategize and use whatever skills he has to win. That assumes a somewhat athletic astronaut, not just some guy who invented a space toilet.
Caveman is used to making his own stuff and living off the land.
All the caveman has to do, is go his own way and wait for the astronaut to starve to death. 
If one has to postulate a fight - then the caveman again. He’s likely to know how to make his own simple weapons from available materials (flint knapping, anyone?).
It is a mistake to think cave men were dumb just because they were more primitive (I’m assuming we are talking homo sapiens all around). All their smarts would have been about using their environment - and that gives them the advantage.
Caveman may have more strength, with all that training the Astronaut may have better stamina, its a toss up.
Assuming they both arrive basically naked, the caveman. Basically living with nothing, he would be used to starting with nothing. If the astronaut is “old school” and out of one of the armed services with basic survival training, he may make it interesting. But I’m thinking it would still be a loss.
Are there coal. sulfur, and diamond deposits in the area?
Look around you – can you form some sort of rudimentary lathe?
Do not underestimate the power of eating right.
Astronaut wins. Every time.
Would they be that poorly nourished though? I wouldn’t like to pit myself against a fully-grown male chimp for example, and they have access to the same resources surely?
I think that in a straight physical fight, the winner is often the one who is more used to fighting, so I’m going for the caveman (assuming, as mentioned above, that it’s a modern science-guy type astronaut, rather than one who has undergone military training).
I would expect the caveman to be more used to violence as a way of life. Sort of, if I don’t kill something I don’t eat.
Caveman
But what if the astronaut is a Ninja and the caveman spends his time as a pirate?
Wins what? The race? The affections of the maiden fair? The bet?
Does the astronaut have a space suit? Does the caveman actually have a cave?
ETA: I strongly recommend you study Skald the Rhymer’s “What if” posts. They represent “best practice” in recreational hypothesizing.
and what if the astronaut’s a woman?
A warp in the evolutionary tree
that’s just crazy talk. What if the astronaut is a unicorn? 