Officers said Ms Michalski was killed during an “intimate encounter” with the suspect, but did not provide any further details.
They managed to establish that the couple met in a nightclub, danced together, and then went to the man’s rented apartment.
He was detained there on Thursday evening and taken into custody.
After questioning, the suspect showed detectives where he disposed of Ms. Michalski’s body, police statement, external said.
It added that the man put the nurse’s body in a wardrobe while he cleaned his flat – in an attempt to “remove traces” of the alleged murder – before driving off into the woods.
Ms. Michalski was reported missing by friends when she did not return to their home.
During the investigation, police said they found evidence that “gives reason to suspect murder.”
After Ms. Michalski’s disappearance, the man made a series of Internet searches, including “what does a dead body smell like after decomposition?”, “how do the police handle missing persons cases?” and “removing the smell of rotting meat.”
It is said that he also looked for whether pigs eat dead bodies, and about the appearance of wild boars on Lake Balaton, a reservoir near Sigliget.
Video footage released by Hungarian police shows a man in handcuffs leading officers through a forest. Service dogs and medical examiners are present.
A Facebook group called Find Mackenzie Michalski says the 31-year-old often went by the nickname Kenzie.
On Saturday, her friends held a candlelight vigil outside the US Embassy in Budapest.
