Greek-born Australian Postecoglou has little time to shape the squad, with a Champions League second qualifying round tie against Danish outfit Midtjylland set to be his first official game in charge next month.
The Hoops turned to the Yokohama F. Marinos head coach after an intended move for former Bournemouth manager Eddie Howe failed to come off.
Postecoglou faces the prospect of lifting a group of players needing a response from last season's shortcomings, as Celtic's bid for an historic 10th consecutive title was left in tatters.
Nicholas - who spent two spells at Parkhead, winning back-to-back championships in 1981 and 1982 in his first stint before joining Arsenal - accepts Postecoglou is up against it even before a ball is kicked.
"The manager has got no time at all. I see this guy as being someone who is already in a really awkward position," Nicholas told the PA news agency.
"I know they're currently looking at different players, but it is getting them in the door - and also getting the ones who don't want to be at Celtic Park out.
"Unfortunately, the ownership tends to wait until they sell before they buy."
Nicholas, 59, added: "He needs to get players in, he needs to get a shape right, and also he's got tremendously difficult away ties (at the start the cinch Premiership season, with trips to Hearts and Rangers in August).
"So it is going to be a testing and difficult time for the new Celtic manager, but it is a challenge.
"If you have not got the players in that you want, I would still expect to see your team having shape, I would expect to see attacking football, so if he gets off to a good start, who know where it could take us?"
Nicholas feels Postecoglou will be given a chance to bed in by the Celtic faithful - provided he sends out a team who give their all for the shirt.
"I think the Celtic fans will be fair to him, but they are sick to death of the ownership at the top level," said Nicholas, who is supporting Prostate Cancer UK's March for Men campaign.
"They want representation and voices heard and they're not being heard, they haven't for a while now.
"If is a slightly slow start for Celtic with the new manager, then I don't see him getting a real awkward time, I think it will all be directed at the ownership.
"Some players have been there for a while and have not been performing well, so they will be getting the flack also.
"The Celtic supporters will accept that they didn't really pick this guy, but he is here. They will support him strongly - but you have to get something out of the team."
:: Charlie Nicholas will be supporting Jeff Stelling as he walks four marathons in four days through the north-east, Yorkshire, Merseyside and London during his fourth March for Men for Prostate Cancer UK later this summer. To find out more, visit https://prostatecanceruk.org/jeffsmarch