Days on an active Major League roster is the term they use, and the Twins have 46 games remaining on their schedule. They can't fit those into 44 days, so even if he never comes to the plate (130 at bats are also a threshold for rookie eligibility) he'll no longer be a rookie next Opening Day.
In the movies the AAA manager calls the player into his office to deliver the good news. If it were me, I would have asked the player his opinion - we're bringing you up to the Show, do you want to come up tomorrow, or wait a few days and preserve rookie eligibility for next year? But that's (just one reason) why I'd be a lousy manager.