Having parted ways with ex-boss Neil Lennon mid-season, Celtic finished the Scottish Premiership campaign in second under caretaker boss John Kennedy, 25 points behind arch rivals Rangers.
The Hoops have roped in former Yokohama F Marinos coach Postecoglou to take over the reins at Celtic Park, a surprise choice having seen talks between primary target Eddie Howe broke down.
Postecoglou does not currently hold a UEFA Pro-Licence, for which Celtic have applied for an exemption, but former Hoops star Caldwell feels the Australian could turn out to be a good appointment.
Caldwell highlighted that Postecoglou has a good track record of winning but stressed Celtic only managing second place would be a disaster.
“I think it is not the appointment that every Celtic fan or even the Celtic board was probably looking for”, Caldwell told Sky Sports News.
“But sometimes these left field appointments can actually be a positive.
“I think he has a very good track record of winning.
“But he is going to find out very quickly he is coming in here into a very different football club where second place is not a failure, it is a disaster.
“And he has to quickly get to grips with that mentality and with Celtic Football Club and what it means to these supporters.”
The Celtic senior squad will return to the club’s Lennoxtown training base on 17th June, but the new boss will likely to be only allowed to join them after a ten-day mandatory quarantine period.