"To see Ole lose his job was tough for me and a lot of people at the club," Carrick told British media on Monday.
"It was a bit of a rush to organise training and speak to the players and carry out the day's work.
"It's a limited time, I'm well aware of that, but it's a challenge I'm relishing at the moment.
"(Villarreal) is a big game for both teams at this stage of the competition, and we can't hide from that. But it's one to go out and enjoy."
United are top of Group F with seven points from four games, level on points with LaLiga side Villarreal, who they beat 2-1 at Old Trafford in September.