Mukai was a fan of the Gundam series in elementary school, which later inspired him to work in the anime industry.[1] He later joined Sunrise and started working on Mobile Suit Gundam SEED Destiny as a production assistant, which was the first series he worked on.[1] Mukai left Sunrise around 2008, and following his departure from the company started working as an episode director. In the early 2010s, he did work for a few Shaft series where he worked with director Akiyuki Shinbo, whom Mukai considers to be a mentor.[1] In 2013, he debuted as series director for the first time with Hyperdimension Neptunia: The Animation.[2] In 2017, Mukai directed the Trickster anime television series.[3]
That same year, Mukai served as episode director for the second season of Blood Blockade Battlefront. While working on the series, he was approached by My Hero Academia producer Yoshihiro Ōyabu to serve as director for the series starting with the fourth season, which Mukai accepted.[4]