The title is a pun on the mountain, Pike's Peak, and on a "piker" or cheapskate.
A small Alpine town is having a festival and the mayor announces a prize of 50,000 kronkheits (a reference to Smith & Dale's famous Dr. Kronkheit sketch) to the first person to climb the Schmatterhorn (a pun on the Matterhorn). None of the townspeople step forward so, dressed in lederhosen with a feather in his cap, Yosemite Sam volunteers and the band plays as he heads up the mountain. Bugs Bunny hears the hubbub and asks about it. When Sam explains the prize, Bugs thinks that would buy a lot of carrots.
Sam climbs a difficult rock face, but Bugs finds a formation nearby that resembles stairs and climbs it, easily passing Sam. Sam grabs Bugs' leg and pulls him down, but also pulls down the boulder from which Bugs was hanging and it flattens Sam. Sam climbs back up to Bug's level and offers to be partners. He climbs ahead and calls to Bugs to throw up his rope. He ties the rope to another boulder and pushes it off the ledge, but it was the wrong rope. The boulder drags Sam down the mountain and all the way back to town. The band plays as he heads up the mountain again.
Sam struggles to pull himself up on a snowy ledge. Bugs gives him a boost, but he goes up and over, sliding through the snow all the way back down through town again. The band plays as he heads up the mountain again.
Sam manages to climb ahead of Bugs and tries to push another boulder down onto the rabbit, but it rolls backwards and smashes Sam instead. Sam climbs back up to Bugs again, but Bugs warns him to be quiet or the snow overhead will become an avalanche. Sam hollers and fires a pistol, but the resulting avalanche carries him away, not Bugs. A St. Bernard digs him out of the snow and pours a drink, but drinks it himself and barks and hiccups back down the mountain.
Sam chases Bugs higher and higher until clouds obscure their vision. When Bugs thinks they've reached the top, Sam pushes him off and begins celebrating. As the camera pans out, wee see he's really somehow on top of the Eiffel Tower, Bugs is safe on a lower level, and the town band at the base of the tower begins playing again.
This is one of the rare Warner Bros. cartoons to have music from the cartoon playing past the iris-out or fade out through to the end card; this was common in MGM cartoons, but rare in Warner Brothers cartoons, as this stopped in 1937 for Merrie Melodies and Looney Tunes. In this case, when the town band plays during the end titles, near the end, it almost sounds identical to a remixed version of the 1938-41 MM end theme played at a different key.
When this cartoon aired on ABC, the part where the St. Bernard makes a martini for himself after rescuing Yosemite Sam from the avalanche was cut. 
"When I'd Yoo-Hoo in the Valley (to My Lulu in the Hills)", uncredited. Written by Henry Russell and Murray Martin
"Little Brown Jug", uncredited. Written by Joseph Winner
"The Bartered Bride, Opening Chorus", uncredited. Written by Bedrich Smetana