Tucker Carlson should be successful in his new role at Fox. He does already have a strong following and seems to be well liked by many. There will always be those who don't care for the "new kid on the block", but overall as long as he does his job well, I think he will be well liked.
Due to her challenging and criticism of president-elect Donald Trump, Megyn Kelly outraged Trump himself and many of his Republican supporters. Carlson's replacement of Kelly should be well-accepted; he occupied a 7 p.m. slot on Fox News in the weeks prior of the announcement and received high ratings for the network.
Tucker Carlson is incapable of being liked, so this outcome is unlikely. First of all, he lacks any semblance of genuineness and appears to be living out a persona as everyone's really annoying cousin during his first year of college. Secondly, Megyn Kelly was able to appeal to wide swaths of Fox's audience because of her beauty and intelligence. Tucker Carlson has neither.
I think Tucker Carlson has an uphill battle ahead of him. Megyn Kelly is a very popular on-air news personality and has a large fan base. Couple that with the reason she left Fox, and the personal courage it took to do so, and you can see why she has endeared herself even more with her fans.