"The boss has placed his trust in me and because of that I have the highest respect for him," Arnautovic told reporters on Wednesday. "I will make sure his trust is not misplaced."
Arnautovic, 24, missed the last three weeks of the Bundesliga season along with club mate Eljero Elia after they were stopped for speeding in separate incidents.
"I'm really looking forward to finally playing again," said Arnautovic. "I will give everything."
He has had a number of disciplinary problems including a brawl with a team-mate during a training session.
Austria, Sweden and Ireland are level on eight points and battling to finish second in European Group C, eight behind leaders Germany.
Sweden, who visit Austria on June 7, have played one game fewer than their two rivals.