Let's say I get into an argument with someone. He falsely accuses me and I deny. He calls the police/local law enforcement of that place and it becomes clear that I was right. Now, what can I do to have him be punished for the time wasted/headache/other possible problems he caused? Can you introduce me to some readings/terms/webpages so I can learn more?
As noted in the comments, the correct answer depends upon where the conduct in question happened.
In most cases, nothing.
There is a qualified privilege from civil liability for reports made to public officials of alleged misconduct. The burden to overcome that qualified privilege is very high.
There may be a minor criminal charge if the police believe that the person making the report intentionally made a false report, but that would ordinarily not result in any compensation to you.
It is unclear from your question what the person making the accusation believed and that is critical to any civil or criminal liability. Simply being wrong is not in and of itself enough to give rise to either civil or criminal legal liability. Someone must know that their accusation is false (or at a minimum recklessly make an accusation without knowing whether it is true or false); if they sincerely believe their accusation, there is no liability even if the accusation arose from their negligence in most cases.
Also, notwithstanding the tag on your post, there would not be a false arrest in this case unless the police did not believe the person accusing you of a crime and arrested you anyway. Police are entitled to take someone accusing someone else of a crime at their word unless there is an obvious reason that they shouldn't believe them.