The new Parkhead boss will kick off his Hoops reign when Danish outfit Midtjylland visit Glasgow on July 20 or 21 before the return trip to the MCH Arena a week later.
But that leaves the former Yokohama F. Marinos head coach with just five weeks to reshape a squad shattered by last year's failed 10-in-a-row bid.
However, 21-year-old defender Welsh is eyeing up next week's return to pre-season training as a chance to wipe the slate clean and prove himself all over again to the Greek-born Australian.
"It's a new manager to impress," Welsh said. "We've got the Champions League qualifier and the first league game away to Hearts, so everyone will be fighting for a place. That all starts next week when we get back into the tough work of pre-season.
"I'm very excited that he's coming in. I've not spoken to him yet but I'm looking forward to meeting up with him.
"It's almost like a fresh start now that we've got a new manager in who plays good football, attacking football.
"We're all excited about the appointment and can't wait to get started.
"He'll be expecting us to come back in good shape and he will be expecting to work with good footballers as well.
"I saw his interview when he joined the club. It was very positive. He said he likes to bring through academy players, which is perfect for me and all the other young boys.
"It's all about getting another chance and trying to prove you're good enough to be here."
The appointment of former Socceroos boss Postecoglou has done little to appease the brewing fury amongst the Celtic faithful following last season's disappointing campaign and their unsuccessful move for Eddie Howe.
The new manager cannot afford to start his tenure by repeating the kind of slip-ups that have seen the club fail to reach the group stages of Europe's premier competition since 2018.
Midtjylland have recent experience of facing Scottish opposition after coming up against Steven Gerrard's Rangers in Europa League qualifying in August 2019, losing 7-3 on aggregate.
But the three-time Danish champions did hammer Slavia Prague - Rangers' conquerors in the Europa League last 16 - 4-1 last season on their way to the Champions League group stages, where they drew with Liverpool.
"Midtjylland will be a quality side," Welsh said. "Anyone who is in the Champions League qualifiers is a top side and we'll have to be very wary of them.
"But we have quality players too and I'm sure it will be a good tie.
"Now the draw has been made, we're very much looking forward to the game.
"The Champions League is where our players want to be - that's where the club wants to play, at the highest level possible - so we're very excited for the two games ahead. Hopefully we do well.
"There's always pressure when you play for Celtic. That's what the club comes with and the standards that have been set over the last few years. Every game we go into, we're expected to win and that will be the same next season."
Meanwhile, Northern Irish champions Linfield could face a showdown with Celtic's conquerors from last year Ferencvaros in the second qualifying round if they can overcome Lithuanians Zalgiris Vilnius.
The Hungarian team, however, still have to see off the winner of the competition's preliminary tournament.
Welsh title winners Connah's Quay Nomads face Alashkert of Armenia in their opening qualifier and have now learned they will take on either Albania's KF Teuta of Sheriff Tiraspol of Moldova if they progress.
And Republic of Ireland champions Shamrock Rovers will head to Switzerland to take on Young Boys if they can edge past Slovan Bratislava of Slovakia in their first-round tie.