Yes you can book a taxi in Shanghai, providing you book it in advance and sometimes not during the rush hours.

Here are the different taxi company's name :
Dazhong Taxi (Turquoise color) : 82222 :
Qiangsheng Taxi (Yellow color) : 62580000
Bashi Taxi (Light Green color) : 6384 1890
Jinjiang Taxi (White color) : 6215 5555

Avoid blue or red color taxis, and thosestarting with a BX number, they are usually either bad or unregistred, and may cheat you on the fare.

Yes, you can do this in probably all the cities. I agree you are gonna have to book in advance at least a week's notice. Most people that book taxis are willing to go travel longer distances so it is worth it to the taxi driver to provide this service. I'm not sure if you are wanting to book it a certain time but I believe many taxi drivers that you book you book for the morning because they usually have a busy schedule in the evenings and afternoons themselves. Everytime I booked it was done in the morning time.