Kingsbury pegged the recovery time for Jones' surgery at three to four months.
Jones had 19 sacks last season and sits at 97 for his career, which is the most in the NFL since he entered the league in 2012.
"He's a tremendous player, as we all know. Great person. Great leader on this team, so it's not going to be easy," Kingsbury said.
After Jones left the field because of the injury, he returned to the sideline and tested his biceps by pushing against a teammate. But Jones was seen grimacing after one push and then sat on the bench holding his arm.
Jones' presence has been a major boon for the Cardinals' defense. The 30-year-old was named to the NFL's all-decade team for 2010-19. Just his being on the field this season has drawn regular double-teams and chips, opening up one-on-one matchups for his teammates.
Jones will be replaced by the rotation of players who have already been getting time at outside linebacker, namely Haason Reddick, Kylie Fitts and Dennis Gardeck, the former undrafted free agent who had two sacks Sunday in the first defensive snaps of his career.
"We have some guys that have seen playing time this year and will continue to play and get more," Kingsbury said. "So it'll be a rotation of guys that we've seen out there which we'll just see out there more."