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It's not exactly explicit, but his reminiscing at that point does imply that he lost to Lance, otherwise he would have been a Champion and THAT would have been trumpeted all over Alola once he came back. But you never hear of him being a Kanto Champion at all, not even from the kahunas. Thus, its easy to infer that he made a great challenge but lost to Lance in the end.
How long ago it was is never specified. Did he begin his challenge right after Alola? Or did he do it while studying to be a Professor? Remember, the MC's mom says that she personally watched Kukui fight and get beat up by the Gym Leaders, although their exact relation isn't quite clear.
