Traveling in china for 5 months, with iPhone 4s. Need Internet and calls. Which company advised?

If you stay in China for such a long time, you need to get a Chinese SIM card and you can choose China Mobile, China Unicom or China Telecom to make local calls. I used China Telecom in China, which is not bad. It is suggested to get a new phone here in China with a new card, which will be much cheaper. A friend of mine did so.
It is hard to tell you how it work, you can ask the shop assistant in the cell phone shop.
If you want to make international phone calls, you had better use Skyper.
If you still study in certain university, please do not worry about the Internet, otherwise you need to pay in advance.
If you want to know more detail, you can refer to https://www.travelchinaguide.com/essential/post_telecom.htm
Hope it is clear for you.

What school are you studying at in China? My son will be going Mid February to study at Sun Yat Sen in Guangzhou, China.