The Potters landed in Ukraine on Wednesday ahead of Thursday night's Group E opener against Dynamo Kiev.
The Staffordshire outfit - who are taking part in major European competition for the first time in 37 years - also face trips to Israel to play Maccabi Tel Aviv and Turkey to take on Besiktas.
That will require the team to clock up far more miles than manager Pulis would like, but the Welshman is determined Stoke will embrace the task of balancing their European commitments with their domestic campaign.
"Someone asked me if I'm excited about being in Europe - I don't think we are in Europe if you look at the teams we are playing!" Pulis said.
"We are in Ukraine, Israel and Turkey. We got the short straw and the travelling distances are horrendous but we are looking forward to it.
"We have to manage the squad; there were some good players not in the team on Saturday [for the Barclays Premier League victory over Liverpool] who will get a chance to show what they can do.
"We play Kiev on Thursday, Sunderland on Sunday, Tottenham on Tuesday [in the Carling Cup] and Manchester United on Saturday - it's not a bad run of games but it is a big call for the next couple of weeks.
"It is what everyone aspires to and we have to cope with, but it is new to us."
Pulis was keen to stress that his words were meant as no insult to Dynamo, a club with considerable European calibre who have former Chelsea striker Andriy Shevchenko among their ranks and knocked Manchester City out of the Europa League last season.
"I've got the greatest respect for all three teams we are playing," Pulis said.
"I know Kiev have played many games in Europe and have more experience than most teams we are going to play - they are a very famous club.
"We watched video of the Man City games from last year, so we know it will be tough.
"Shevchenko is a great player with fantastic pedigree. He has played in England and knows what English football is all about, so of course, whether he starts or comes on as a substitute, we will give him respect."
Stoke's squad is being stretched by their foray into Europe this term and several players have not travelled to Ukraine, including record signing Peter Crouch.
Pulis is likely to give a run-out to Crouch's fellow new recruits Cameron Jerome - who is yet to make his Potters debut - and Wilson Palacios, while Thomas Sorensen will take the rested Asmir Begovic's place in goal.
The 1-0 win over Liverpool - which extended Stoke's unbeaten start to the season to eight games in all competitions - was watched by Dynamo coach Yuri Semin, who is well aware of the strengths Pulis' team possess.
"They are a very dangerous and difficult side with physically strong players and a well-organised defence," Semin told www.fcdynamo.kiev.ua.
"They are also very good at set-pieces, so their positive results in this season's championship are absolutely logical."