Several options here. First is those Japanese clubs (with girls). They usually have a good selection. There are about 200 in Shanghai.
Otherwise, for pure karaoke, go to Big Echo (yes, the Japanese company). There are now 3 locations in Shanghai -- check http://www.bigecho.com.cn/cn/index/
I just went to Big Echo last weekend. It was nice, with a big selection of Japanese songs. Somehow some of the songs didn't come up -- something wrong with their system.