The Parkhead defender reckons that Ronny Deila's side will have to be at their best to avoid a slip up in Iceland on July 15 for the first leg of the second qualifying round tie.
The champions head out to Austria this week to begin an intensive pre-season training camp - with four games in eight days - which new boss Deila hopes will have Celts ready for the real thing when they step out in Reykjavik.
KR are already well into the Icelandic season and Celtic discovered in last season's qualifying trail how important that was to Swedish side Elfsborg, as the Hoops only edged the tie 1-0
The fact that Celtic almost suffered a huge scare against Shakhter Karagandy, overturning a 2-0 defeat in Kazakhstan before squeezing through into the group with a 3-0 victory in Glasgow, is all the proof Mulgrew needs to take KR seriously.
'I've heard of them but don't really know that much about them," said Mulgrew. "We know it will be hard. Every team is scrapping to get into the Champions League. They are nine games into their league so it will be difficult.
"We just need to try and match them. Last year against Elfsborg, they were into their season and you could see playing the game how much they wanted to be in the Champions League."
He went on: "You need to be prepared for it and we'll need to get up to sharpness as quickly as possible and I'm sure the manager will have us ready.
"We're experienced in Europe now. It's important we take on the new manager's ideas as quickly as possible. That's what we're working on every day, twice a day, having meetings and trying to get the message across.
"We've been in the Champions League group stage the last two years and we want to get there again. But it's not easy. Some people maybe forget how hard it is to get there through the qualifiers, these teams are into their season and it's difficult.
"Going to Karagandy last year was a difficult game. That's the type of games you come up against, a hostile atmosphere. Every team is scrapping to get there. It doesn't matter about ability when a team wants it and the prize at the end is so big and makes it very difficult.
"Even after qualifying for the last 16 in 2012-13, to go back to not doing it last season was so disappointing, even though we had played in the group stages. So we've got to push towards that and we've all got to get behind it and hopefully we can do it."