There is nothing wrong with the first sentence. It just expresses a different meaning than the second.
"Julie might like to visit Kyoto" indicates that Julie hasn't yet decided if she is going to plan the visit to Kyoto. She isn't sure if she wants to go. "Might" could also indicate that her decision to visit Kyoto depends on some other factor that we don't know about. "Julie might like to visit Kyoto, but if the weather is bad, she will stay in Tokyo."
"Julie would like to visit Kyoto" means that she definitely wants to go to Kyoto, but the trip has not been planned yet. There is still a chance that she won't go, but if you give Julie the choice, she will choose to visit Kyoto. It could also express that she wants to go to Kyoto, but cannot. "Julie would like to visit Kyoto, but train tickets are too expensive."
The phrases are very similar. They just express different degrees of how much Julie wants to go.
"Julie might like to go to Kyoto if the train tickets are not too expensive."
Julie has a little bit of interest in going to Kyoto, but it's not worth it to her if the train is expensive.
"Julie would like to go to Kyoto if the train tickets are not too expensive."
Kyoto is definitely on Julie's "places to visit" list, but she doesn't have a lot of money right now and might have to wait until a later trip to Japan to make it happen.