Since Brackman, unlike the other two, is on the team's 40-man roster, he was officially recalled and will be in uniform on Thursday, although he said he was not planning on pitching in the Major Leagues this season.
Montero and Betances, on the other hand, will watch the game from the stands after participating in the pregame routine.
"I'm not really expecting much of anything. I just want to be here and soak it up," said Brackman. "I don't expect to pitch, but I'd love to be in the dugout."
General manager Brian Cashman did say that Brackman would be available in the bullpen if manager Joe Girardi wants to use him.
Brackman, a 24-year-old, 6-foot-10 right-hander, was 10-11 with a 3.90 ERA in 27 games (26 starts) between high Class A Tampa and Double-A Trenton.
Betances is another tall right-hander, coming in at 6-foot-8. The 22-year-old went 8-1 with a 2.11 ERA in 17 starts between Tampa and Trenton.
"It's an honor to be here," said Betances, who was born in Brooklyn and used to attend games at the old Yankee Stadium. "I never thought I'd be in this situation. It's a wonderful feeling."
Montero, who doesn't turn 21 until the end of November, hit .289 with 21 homers and 75 RBIs for Triple-A Scranton/Wilkes-Barre. The catcher of the future hopes he's here for the long haul.
"For four years working in the Minors, I'm finally here. Now I want to work hard to stay here all the time," Montero said.
Bringing Minor League prospects into the clubhouse without dressing them in uniform isn't uncommon for the Yankees this time in September, including with Phil Hughes.
"[It's] just to have a feeling what it's like up here, the atmosphere," Girardi said. "Simple things like getting familiar with the clubhouse, where everything's at and how it works around here."
Betances admitted to getting lost a few times already in his first trip to the new Yankee Stadium, while Brackman couldn't get over the magnitude of the park.
"It blew me away," he said. "When I walked out on the field for the first time, it was amazing how big the place is. Everything is so nice and first class. It's pretty much what I thought it would be."