Born and raised in San Antonio, Texas, Jung attended Douglas MacArthur High School, where he played baseball[1] and football.

[2] Undrafted out of high school in the 2016 MLB draft, he enrolled at Texas Tech University to play college baseball for the Red Raiders.

[7] That summer, he played in the California Collegiate League[8] where he hit .368 with two home runs, 31 RBIs, and a .454 on-base percentage over 117 at-bats.

[18] He was named the 2019 Big 12 Co-Player of the Year (along with Baylor's Davis Wendzel) after batting .333 with nine home runs and 49 RBIs through 49 games.

[20][21][22] Jung was the recipient of the 2019 Bobby Bragan Collegiate Slugger Award, which honors the top Division I baseball player within Texas, Arkansas, Louisiana, New Mexico and Oklahoma each season.

[28] On July 15, Jung was promoted to the Hickory Crawdads of the Class A South Atlantic League and hit .287/.363/.389/.752 with one home run and 23 RBIs over forty games for them.

[31] After hitting .308/.366/.544 with ten home runs and forty RBIs over 43 games, he was promoted to the Round Rock Express of the Triple-A West in mid-August.

He made his MLB debut that night and in his very first at-bat, he hit a home run off of Ross Stripling of the Toronto Blue Jays.

After playing in just four games, Jung suffered a fractured wrist following a hit by pitch, and was ruled out for 8–10 weeks after undergoing surgery.