The two-day auction, which ends on Saturday, includes more than 800 items owned by Starr and his wife, actress Barbara Bach.
Starr, now 75, used the kit in more than 200 performances between May 1963 and February 1964. He also played it on recordings including Can’t Buy Me Love, She Loves You, All My Loving and I Want To Hold Your Hand.
A portion of the auction proceeds will go to the couple’s Lotus Children Foundation, which focuses on world social welfare issues.
Mr Irsay owns prominent guitars from the group’s other three members – Paul McCartney, the late George Harrison and the late John Lennon. Earlier this year he also purchased a drumhead that will fit the set he just bought.
“Finally, after 45 years, the Beatles are together again,” Mr Irsay said.
Mr Irsay is a huge music fan and an avid collector. He owns about 30 famous guitars, including those used by Bob Dylan, Jerry Garcia and Eric Clapton.