Born in Christchurch, Armstrong began karting in 2010, winning all of his major domestic titles in New Zealand from 2011 to 2014. His highest placement at the Karting World Championship was twelfth in the KF2 category in 2015. Motorsport.com ranked him third among the most exciting karting talents of 2016.
Armstrong signed with the Ferrari Driver Academy in December 2016, alongside campaigns in Italian F4 and ADAC Formula 4 with Prema Powerteam in 2017. He won the Italian F4 Championship that year, clinching the title with two races to spare. In ADAC F4 he finished runner-up to Juri Vips by 4.5 points.
In 2018 Armstrong graduated to the FIA Formula 3 European Championship with Prema, finishing fifth overall. The series merged into the inaugural FIA Formula 3 Championship in 2019, and Armstrong emerged as runner-up to Prema teammate Robert Shwartzman, collecting three victories, seven podiums, three poles, and four fastest laps across the season. He dedicated his race two win at Spa-Francorchamps to Anthoine Hubert, who died in a crash the night before.
Armstrong also contested the Toyota Racing Series in New Zealand for multiple seasons, reaching runner-up in 2019 behind fellow Kiwi Liam Lawson.
Armstrong moved to the FIA Formula 2 Championship in 2020 with ART Grand Prix, partnering Christian Lundgaard. He marked his debut with a second-place finish in the opening Austrian feature race, capitalising on problems for rivals ahead. Despite that strong start, his season became inconsistent and he ended thirteenth with 52 points, far behind Lundgaard's 149.
He joined DAMS in 2021 alongside Roy Nissany. The season contained moments of pace — Armstrong won the Jeddah sprint race, his first Formula 2 win — but mechanical misfortune and racing incidents kept him outside the upper reaches of the standings. He ended thirteenth again with 49 points.
For 2022 Armstrong moved to Hitech Grand Prix alongside Vips. He claimed victories at Imola, the Red Bull Ring, and Zandvoort during the season and improved to 93 points, but finished thirteenth in the standings for the third consecutive year.
Armstrong tested an IndyCar for the first time with Dale Coyne Racing at Sebring in October 2022. He was signed by Chip Ganassi Racing for 2023, initially on road courses and street circuits. He finished eleventh on debut in St. Petersburg, achieved multiple top-ten results including seventh in Toronto, and ended the season twentieth overall with 214 points. He was awarded the IndyCar Rookie of the Year.
Armstrong extended his Chip Ganassi Racing contract for the full 2024 season. He suffered an early engine failure in his debut Indianapolis 500, but secured his first IndyCar podium with third place at Detroit and added four more top-ten finishes to end fourteenth in the standings with 298 points.
For 2025 Armstrong moved to Meyer Shank Racing, partnering Felix Rosenqvist. Despite a retirement in the St. Petersburg opener, he posted strong results including seventh at Thermal and in the Sonsio Grand Prix. He secured his second career IndyCar podium in the second Iowa race and closed the season with consistent top-ten results, ending eighth in the standings with 364 points — eight behind teammate Rosenqvist. He extended his contract with Meyer Shank Racing for 2026.
Armstrong formally parted ways with the Ferrari Driver Academy after his 2021 season, later describing it as "a dead end." He had conducted F1 test sessions at Fiorano in the SF71H, completing more than 200 km across two separate test days in 2018.
Armstrong hosts a motorsport and food podcast, Screaming Meals, alongside James Harvey Blair, Clement Novalak, and Callum Ilott. Growing up, his favourite driver was 2007 Formula One world champion Kimi Raikkonen.
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