"I bought a Shinkansen ticket."
For your reference:
The Shinkansen network: 「しんかんせん」 : 「新幹線」 is a network of high speed trains in Japan. Shinkansen literally means "new trunk line" in Japanese.
They are more commonly known as "bullet trains" : "dangan ressha" : 「弾丸列車」 : 「だんがんれっしや」 , with speeds of upto 320 kilometres per hour.
Though particles are often dropped in conversation, it is not common to say Shinkansen Kippu(新幹線切符). Shinkansen no Kippu(新幹線の切符) is much better.
In Japanese class we call the しんかんせん the bullet train. Not sure why Duolingo doesn't use that as a translation.
Personally, I'd always use きっぷ for trains and チケット for something like a movie.