There plans out there, mostly on contracts with yearly payment or monthly fee, that allow you to receive your calls free of charge. Nevertheless, one should remember that the main focus should be on reducing your total mobile phone bill at the end of the month, especially if you also call a lot.
The plan I am currently on, with China Mobile, has a monthly fee of 60 yuans, which gives you 360 free minutes. That means within those first 360 minutes you will be paying 0.167yuans per minute instead of the regular 0.4 or 0.6. If I use more than 360mn, then I am charged 0.2/mn for any additional minutes, still rather cheap by the market standards. There are several types of those plans, that you can choose from depending of your average monthly calling time. The drawback is, if I don't call up to the 360mn, I am still charged the 60 yuans. And the free minutes are only for local calls, meaning calls you receive when in Shanghai or calls you make to Shanghai numbers. Rooming charges are the same as the standard plans.
China Unicom has a card that allows you to receive calls for free, the monthly charge is RMB10 but usually the voice quality is considered lower than China Mobile.
If you want to switch to CDMA, of China Unicom too, the charges are usually much lower, for receiving if I remember well it is something like 0.02 so almost free, and the voice quality is even better than GSM.
It is always better to receive calls on land phones, or if you are very mobile, you can get a Xiaolingtong, these are type of mobile phones with land phone numbers (not 130/133/135/138 etc. but instead the regular 021+ 8 numbers), the reception is free and for calling you are charged as a land phone. The Xiaolingtong phones themselves are cheap (you can get a good one for less than 500RMB, including the number). But once again network coverage is rather poor and you can sometimes experience loss of signal. China Mobile, China Unicom and even China Telecom and China Netcom (fixed line operators) have their own Xiaolingtong.
About voicemail, songs downloading or other options, the best thing to do is to go to a service center of your operator with a friend and tell them what you need, usually they are very helpful. I couldn't find an english version of china mobile and china unicom websites so it will be rather difficult for you to gather enough information and do it yourself.
