"The muzzle hit his neck and stomach and he was taken to the hospital but he did not survive," said police officer Chaina Ram.
Lampe, 29, is from Argentina and moved to Wellington in 2009. He worked at Weta Digital as a senior compositor.
"The couple came to our hotel on Saturday from Pushkar and that same day this strange accident happened, they were supposed to stay here for two days before heading to Agra," said Kunal Sonar, of the Hotel Mustache, where Lampe was staying.
Tributes for Lampe have been posted on social media.
Francisco Espeche said Lampe was a fan of the Lord of the Rings movies and ended up joining the Oscar-wining Wellington special effects company that made them.
"Great guy and always with a smile."
Lampe would have turned 30 this weekend and he and his fiancee had planned to get married next month.
Lampe had trained at the Image and Sound Design at the University of Buenos Aires. He met his fiancee Paula in Wellington.
In India it is common for bulls and cows to move freely through the streets.
Facing heavy criticism after Lampe's death, the Jaipur Municipal Corporation has started a drive to control stray animals in the city, the Hindustan Times said.