Lex having hair isn't a problem. His personality doesn't fit the character in the least bit. Not from what I saw anyway, or how he acts in general.
How exactly? Maybe it doesn't fit any movie version of Lex we've seen, but it fits the Byrne/Morrison Lex Luthor perfectly. The thing is that we have to understand what exactly makes the character of Lex Luthor. I'm going to paste part of a post I've made about him somewhere else, partly because it's relevant and partly because I really love the character:
Look at him: the prime human being, a true genius with a massive appreciation for science, self-made billionaire, top physical shape. The pinnacle of everything manking can achieve. The moral and ethical superman who will not be chained by anything other than his will and his ambition.
Then comes this benevolent alien who comes to act as humanity's babysitter, taking the title that should be rightfully his. And then Lex starts to obssess over him. He believes Superman is hindering humanity's real potential. He's also jealous at Superman for being far more successful and recognized with much less effort. He dedicates his life to defeating him, not realizing how much good he could do if he put his intellect towards something else. And that is Luthor's downfall. By all rights, he could propel humanity to immeasurable lengths, but instead he's a megalomaniac. He projects his own personality on Superman and thinks that Superman views himself as superior to others because of his powers just like Luthor does.
This is why he hates Superman. Kal-El's presence makes Luthor simply a normal man. Luthor's propaganda that Superman's presence spells the end to human ingenuity and attainment is really his own insecurity that he is no longer the best person on earth because Superman exists.
I find one of the most defining moments of the character is, appropriately, in the best Superman story told so far: Grant Morrison's All-Star Superman, specifically, issue five: The Gospel According To Lex Luthor. This issue is notable for not featuring Superman at all, which turns him into more of an idea than an individual. More importantly, it focuses on Lex’s point of view, an entirely different notion of the Man of Tomorrow and the entire motivation of Luthor as an antagonist.
Luthor raises the question of what the world would be like without Superman. We’ve already seen how Superman has inspired people to be better, in all aspects of life. For them, Superman is a model to aspire to. In Luthor, we see the opposite of that: what would happen if a man could dedicate his entire life to be the best, both physically and mentally, but still be unable to match up to Superman. That would be incredibly frustrating, and this issue does a good job of conveying that. He uses Clark Kent himself as an example, saying he would never have Lois as long as Supes is around. How would the rest of humanity even begin to compete with Superman?
This is a short, but fantastic speech, because it summarizes the entire conflict between the two characters. Lex can’t handle the competition. It’s a distinctly human motivation, this inability to accept that someone better than you is out there. This is why Luthor is the greatest Superman villain, despite being a lesser threat in comparison to the likes of Darkseid and Brainiac: he is the perfect foil to the hero. He exists only to defeat him. He's completely capable of being the greatest man to have ever lived, if he stopped focusing so much on being the hero that never was.
Lex Luthor doesn't need to be strong, or talk tough, or look like your traditional definition of a powerful villain. What makes him is his intellect, his intentions, his ideology. Jesse Eisenberg, from the little we've seen, has managed to capture the elegance and superiority that Lex requires. Not only that but he is a talented actor who will, without a doubt, bring us the serious Luthor portrayal we have been waiting for.
