Cooney did not receive any scholarship offers after his season at Bridgton, but was given a chance to make the Syracuse University football team as a walk-on.[3] He made the team, but did not play in his first two years (2016, 2017).[2] He saw his first collegiate action in 2018, serving as the holder for field goals and extra points.[2] Cooney continued in this role for the 2019 season, his junior year.[2]
As a fifth-year senior in 2020, Cooney received his first action as Syracuse punter.[4] He recorded 74 punts averaging 44.8 yards, leading all of the FBS, and had 24 kicks downed inside the 20 compared to just three touchbacks.[4] Cooney was an honorable mention All-American according to Pro Football Focus (PFF) and was a third-team all-conference selection.[2] He was also named the winner of the Brian Piccolo Award, given to those the "most courageous player in the league," for overcoming his battle with cancer in high school.[2]
In January 2022, after spending the entirety of the 2021 season as a free agent, Cooney was signed to a future contract by the Arizona Cardinals.[8] He was released on August 23.[9]
Indianapolis Colts
Shortly after being released by Arizona, Cooney had a tryout with the Indianapolis Colts.[10] Although not signed at first, Cooney was brought to the Colts' practice squad on October 25, 2022, following a poor performance by the team's punter Matt Haack.[11] After Haack performed better in the following game, Cooney was released on October 31.[12]
Arizona Cardinals (second stint)
After Andy Lee was ruled questionable with an illness in Week 9, Cooney was re-signed by the Arizona Cardinals on November 5 to the practice squad and elevated to the active roster for their match with the Seattle Seahawks.[13] He was released on November 9.[14] He signed a reserve/future contract on January 11, 2023.[15] He made the final roster by winning the position battle over Matt Haack.[16] On October 3, after four games played, in which he recorded a total of 13 punts for 592 yards (a 45.5 average), he was released in favor of Blake Gillikin.[17][18][19]