The former United and England midfielder was appointed this week on a deal to the end of the season, but faces a baptism of fire at Old Trafford when local rivals Manchester City visit on Saturday, seeking to boost their Premier League title hopes.
Far from challenging for the title, seventh-placed United are 17 points adrift of top spot.
For just the second time since the 1989/90 campaign, the 20-time English champions failed to qualify for Europe this season and are far from guaranteed a return to continental competition next term.
"We want to be near the top of the league, we want to be top of the league. That's pretty obvious to say," Carrick told a pre-match press conference on Friday.
"But we've got to take some small steps towards that and European football obviously would be a step forward.
"I'm eager to succeed. We've got a big job to do. It's an important job and it always is here about winning games, to play well and to do it in a certain way."