Hi Arilou1980,
Try some of the ones below which are from 5 to 15 mins from Times Square. Not all have one bed apartments but you will get something for around your budget.
Can I also suggest that just before you arrive, have a good search on the Cityspace on this webbsite and the others and contact those with photos. Good ones do go quickly so if you really like something, have a look at it adn put an offer in on it. You can always pull out if you find something better. We've got some helpful Advice on Leasing on our webbsite with some tips for you.
Lao Shimen Shinyuen & Huangpu Zhong Xin at Xi Zhang/Fuxing Rds
Zhong Kai on Da Gu Rd/Wei Hai Rds
Dong Huai Hai Gongyu at Huai Hai/Renmen Rds
Jia Li Yuen at Dong Hu/Huai Hai Zhong Rd
Sifang Xin Cheng on Ju Lu Lu/Shan Xi Nan Lu
Dong Fang Bail at Xiangyang Rd/Huai Hai Zhong Rds
Good luck!
Adrienne
http://www.shanghaiprops.com