Watch him well and do the same thing.
Watch him closely and do likewise.
Watch him carefully is much closer to the meaning of 彼をよく見て. “Watch closely” in English sometimes has the connotation “keeping an eye on the development or the change in situation for a certain period”. But よく見る in Japanese merely means “pay attention and try to understand when you watch something”. The target is not necessarily changing. よく見る even can be done with a quick yet careful look.