Stark played fair with the Parkhead club by taking Mulgrew off at half-time, ensuring he remained fit and available for Saturday's game at Aberdeen and next week's Champions League tie at Benfica.
And Mulgrew praised Stark, who did his claims to be appointed full-time boss no harm by steering Scotland to a narrow friendly win via two early Jordan Rhodes goals.
"He was under a bit of pressure last night and I'm sure he wanted to step in and do a good job," said Mulgrew after winning his fourth cap.
"He did that, all the boys were impressed with how he went about training and preparing the team.
"I have been really impressed by how Billy Stark has gone about his business this week. He has done things really well and been very professional.
"I worked with Billy before at Under-21 level so everything is always very professional.
"We all want a new manager appointed as soon as possible, but at the same time we want it done right. I'm sure the SFA will do that and choose the right person to do the job for Scotland.
"I was sorry to see Craig Levein go, he's a good manager and he gave me my debut so I have a lot to thank him for. It was unfortunate that some bad luck counted against him. But that happens in football and I'm sure Craig will be stronger because of that."
Mulgrew said Rhodes' two goals had been the highlight of a sound Scotland performance.
"Jordan has been banging goals in down south for a few years now so it's great that he is playing for Scotland and hopefully he will keep scoring for us," he said.
"These kind of games can be tricky, playing away from home against a team that is always going to be up for it. You can see they were putting in some hard tackles and putting themselves about a bit."
Mulgrew played in a slightly unusual central midfield role alongside Darren Fletcher but – as the only Celtic man in the side – he came off at half-time as pre-arranged.
He said: "I got a lot of the ball and like playing in that position.
"It was arranged before the game so, yeah, I'm fine. Celtic have a big couple of games coming up, but I'm fit and raring to go."