Cease was a student at Milton High School. His fastball velocity peaked at 98 miles per hour (158 km/h), although it averaged 91 to 95 miles per hour (146 to 153 km/h) when he pitched for the school’s baseball team. When Cease did not pitch, he filled in at shortstop. During his junior year (2013), Milton won the baseball state championship and he went 9–0 with a 0.81 ERA in 69.1 innings pitched with 100 strikeouts. Additionally, Cease participated in a second consecutive state championship series, but Lambert High School prevailed. Cease participated in summer competitions for Team Elite, based in Winder, Georgia, and did well enough to receive invitations to the Perfect Game All-American Classic and Perfect Game National Showcase. Cease pledged to play baseball for the Vanderbilt Commodores and attend Vanderbilt University on a baseball scholarship.

Before Cease’s senior year at Milton in August of 2013, Jonathan Mayo of MLB.com thought Cease might be selected in the first round of the 2014 MLB draft. On March 3, 2014, during his senior season, Cease departed a game due to elbow soreness. Upon further examination, it was determined that he had partially torn the ulnar collateral ligament (UCL) in his pitching arm.

Due to his injuries, Cease dropped out of school and was chosen by the Chicago Cubs in the sixth round of the 2014 draft. With a signing bonus of $1.5 million, he signed with the Cubs, far more than the $269,500 suggested for his position. He had Tommy John surgery to repair the UCL after signing.

In May 2015, he started throwing again and was a pitcher for the Rookie-level Arizona Cubs. Batters have hit.145 against him in 11 games (eight starts). Cease was ranked as the second-best prospect in the Arizona League by Baseball America at the conclusion of the 2015 campaign.