Olson attended University High School in Spokane Valley, Washington. He enrolled at Gonzaga University and pitched for the Gonzaga Bulldogs baseball team. He pitched for the Bulldogs as a freshman in 2009, but took a redshirt in 2010 when he suffered a shoulder subluxation. In 2012, his junior year, he was added to the Bulldogs' starting rotation, and pitched 110 innings.

The Oakland Athletics selected Olson in the 17th round of the 2012 Major League Baseball draft. Olson did not sign with Oakland, returning to Gonzaga for his senior year. As a senior, Olson had a 9-4 win–loss record and a 2.48 earned run average (ERA) in 101 ²⁄₃ innings pitched. He was named the West Coast Conference's Pitcher of the Year. The Seattle Mariners selected Olson in the seventh round, with the 207th overall selection, of the 2013 MLB Draft. He signed with the Mariners and began his professional career with the Everett Aquasox of the Class A-Short Season Northwest League. He started the 2014 season with the High Desert Mavericks of the Class A-Advanced California League, and was promoted to the Jackson Generals of the Class AA Southern League during the season.