Hollins was born in Arizona on November 8, 1991 and moved to Memphis, Tennessee in sixth grade. As a child, Hollins would practice basketball for hours until his father forced him to leave.
Hollins attended Germantown High School, where he lettered on the basketball team under coach Newton Mealer. He averaged just under 16 points per game as a junior playing alongside future Belmont signee Ian Clark on a team that went 20-12. Hollins drew attention from college programs including Arizona State, Arkansas, Baylor, Georgia, Minnesota, Oregon State and the University of Memphis. As a senior, he averaged 18.9 points per game in leading Germantown to a 24-9 record. The team reached the Region 8-AAA quarterfinals, losing to state champion Melrose High School. The Memphis Commercial Appeal named Hollins to the Best of the Preps Class AAA Team. Despite sharing a last name, growing up in the same part of Tennessee, and wearing the same No. 20 jerseys in high school, Austin and Minnesota teammate Andre Hollins are not related.
As a freshman at Minnesota playing under coach Tubby Smith, Hollins averaged 4.5 points and 1.5 rebounds per game. He started five games and posted a 13-point performance against Indiana. After the team failed to receive an invitation to the NIT, Hollins considered transferring.
In his junior year, he became recognized for his stifling defense. "I take a lot of pride in that and it's kind of fun, locking down another team's top guy ... you may not hold them to zero points, but just making it tough for them -- it's fun," said Hollins. In the NCAA Tournament, Hollins contributed 16 points, four steals and a career-high seven assists against UCLA in the Round of 64. He tallied 10 points and three rebounds in the season-ending loss to Florida in the Round of 32.
Hollins scored 20 points and grabbed a career-high 14 rebounds in a win over South Dakota State, in addition to eclipsing the 1000 point mark for his career. As a result, he was named Big Ten Player of the Week for the week of December 16, 2013. In his senior year, coach Richard Pitino referred to Hollins as "a coach's dream" for his strong work ethic and lead-by-example style. He endured a cold spell in the middle of the season but regained his form towards the end of the regular season, scoring 27 points in a win over the Iowa Hawkeyes. In the quarterfinals of the 2014 National Invitation Tournament, he went 6-11 from behind the arc and scored a career-high 32 points to help defeat Southern Miss. He scored 12 straight points in the first half to help Minnesota retake the lead. Minnesota won the NIT by defeating SMU 63-65 behind 19 points from Hollins. He made the go-ahead 3-pointer with 46.1 seconds remaining and was named NIT most outstanding player. For his career, Hollins played in a school-record 140 games, breaking the record previously held by former teammate Rodney Williams. He led the team in steals with 75 and finished second in scoring average with 12.4 per game.
Hollins went undrafted in the 2014 NBA draft. On August 2, 2014, he signed a one-year deal with Denain of the LNB Pro B. In 34 games for Denain, he averaged 8.4 points and 1.7 rebounds per game.
On June 20, 2015, Hollins was locked in to play for the Brooklyn Nets in 2015 NBA Summer League.
On July 17, 2016, he signed with Finnish team Kauhajoen Karhu.
Hollins is the son of Angela and Lionel Hollins. He has three siblings: Lamont, Jackie and Anthony. His father is a former NBA player and coach who won an NBA championship with the Portland Trailblazers in 1977. He last coached the NBA's Brooklyn Nets.