Mullen joined the Ballyhale Shamrocks club at a young age and played in all grades at juvenile and underage levels. He enjoyed championship success at minor level before later winning back-to-back championships with the under-21 team. Mullen made his first senior championship appearance on 17 September 2017 in a first round game against Rower-Inistioge. He scored a point from play in the 2-16 to 0-12 victory. On 28 October 2018, Mullen scored a point from right wing-forward when Ballyhale Shamrocks defeated Bennettsbridge by 2-20 to 2-17 to win the Kilkenny Championship. He lined out in the same position when Ballyhale faced Ballyboden St. Enda's in the Leinster final. Mullen scored 2-01 from play in the 2-21 to 0-11 victory. He was switched to the left wing-forward position for the All-Ireland final against St. Thomas's on 17 March 2019. Mullen scored five points from play and collected a winners' medal following the 2-28 to 2-11 victory. He ended the season by being named at right wing-forward on the Club Hurling Team of the Year while he was also named Club Hurler of the Year. On 27 October 2019, Mullen lined out at right wing-forward when Ballyhale Shamrocks faced James Stephens in the Kilkenny Championship final. He scored three points from play and collected a second successive winners' medal after the 2-21 to 1-15 victory. On 1 December 2019, Mullen was again selected at right wing-forward when Ballyhale Shamrocks faced St. Mullin's in the Leinster final. He scored three points from play and collected a second successive provincial winners' medal following the 1-21 to 0-15 victory. Mullen first lined out for Kilkenny as a member of the minor team during the 2016 Leinster Championship. He made his first appearance on 14 May 2016 and scored eight points from right corner-forward in a 2-18 to 1-19 defeat by Dublin. Mullen was eligible for the minor grade again the following year. On 2 July 2017, he top scored with 1-09 when Kilkenny defeated Dublin by 3-15 to 1-17 to win the Leinster Championship. Mullen ended the season by being included on the inaugural GAA Minor Star Awards Hurling Team of the Year. On 20 June 2018, Mullen made his first appearance for the Kilkenny under-21 team. He scored two points from left wing-forward in a 1-17 to 3-13 defeat by Galway in the Leinster Championship. On 24 June 2019, Mullen made his first appearance for Kilkenny's inaugural under-20 team. He scored two goals from left wing-forward in the 1-20 to 0-16 defeat of Laois. On 17 July 2019, Mullen won a Leinster Championship medal as an unused substitute when Kilkenny defeated Wexford by 1-17 to 0-18 in the final. Mullen was drafted onto the Kilkenny senior team prior to the start of the 2019 Leinster Championship. He made his first appearance on 11 May 2019 when he lined out at right wing-forward in a 2-23 to 1-21 defeat of Dublin. On 30 June 2019, Mullen scored three points from left wing-forward when Kilkenny suffered a 1-23 to 0-23 defeat by Wexford in the Leinster final. Mullen was selected at right corner-forward for the All-Ireland final against Tipperary on 18 August 2019, however, his participation was in doubt after being hospitalised with an illness in the week leading up to the match. He was declared fit to play but was held scoreless in the 3-25 to 0-20 defeat. Mullen ended the season by being shortlisted for the Young Hurler of the Year award. On 24 February 2020, it was confirmed that Mullen would miss the rest of the season having torn a knee cruciate ligament in Kilkenny's draw with Clare in Round 4 of the National League. In 2021, Mullen was named as captain of the Kilkenny team for the 2021 hurling season. Adrian is an All Star and a former Young Hurler of the Year.