Disney is King in Japan - movies, merchandise, and theme parks. It's a distant number two, but Universal Studios Japan gets the silver medal for popularity in the country of cute.
One thing about USJ and several other theme parks in Japan to make note of: lots of people come early to see the shows and ride the rides, but a second wave of admissions happens when the park's "night time" discount kicks in (this may not be available anymore), and by about 5 PM the park was pretty crowded. I had more experiences to have anyway, and I saw and did everything I intended, so I bid the park adieu.
Universal Studios Japan will have a Harry Potter world with another family roller coaster and Harry Potter and the Forbidden Journey, taken from the US parks. This portion of the park opens in mid-July. Expect massive crowds, especially because it's right in the middle of summer vacation!
Tickets at the time of this writing are around 7000 yen for one day, 12,000 yen for two days. There are "front of the line" passes for 4 or 7 rides (~3000 and 5000 yen additional, respectively). The park has varied hours based on how busy it expects to be. Access is via JR's Universal City Station on the Yumesaki Line; some trains on the Loop Line from Osaka Station go to the park (about 10 minutes), or transfer at Nishikujo Station (5 minutes).