I know this is a bit of a strange question about anarchism, but how would justice be administered in such a society? Would men have to arm themselves to protect what's theirs? Would the justice against a murderer be doled out by a court or by a group of private avengers?

Yes. Ideally, people should to have conscious to so as not to commit really terrible crimes, but just in case there is public outrage and lynching, I think.

Just in some situations, the court is not always fair

I think more often than not the courts aren't fair but the intimidation of dealing with the State is enough to keep people in line. On the one hand, police understand that strength keeps criminals in check and litigation does not. On the other hand, innocent people can fall victim to the bully boys of the State.

So you're left with trying to hope that people in an anarchist society do what's morally right- and human nature says we can't do that- or trying to legislate morality, which defeats the purpose of anarchism. The only way anarchism could give justice is to make every man a vigilante, which would turn into a bloodbath.

In fact, it depends on the civilizedness of the people themselves. I hope that one day a time will come when a person will have enough self-control to not need a "daddy with a punishment belt" represented by State. In current time, humanity is not ready for anarchy.
However, there are a few historical examples the lands on which there was anarchy and there was no bloodshed

I would think that the only function of government would be to arbitrate judicial cases. A judge could be appointed to do that and serve as General of the armed forces in times of war. That would be all they would do.

Hm, may be, may be)