In terms of price you can expect, it depends on your standards really. You could pay as little as a couple of thousand RMB per month or over 10,000 RMB per month. If you want to bargain you will likely need to be in Nanjing for that and would need at least a few weeks at best to sort the place out.
You can find real estate listings in the classifieds on this site, and I would recommend putting up an "apartment wanted" posting there as they are quite visible.
If you have to do it all from overseas I would recommend using an agent you can trust. I have had good experiences with both James Wang at CMR and Frank Pan at Sun Homes.
I know Vinski Solutions specialise in English systems. But if your Chinese is good on the 2nd and 3rd floors of Bai Nao Hui you can find computer repair shops.
Try searching Hertz in our classified section. The made some classified ads some while back and I am sure they can do daily rental.
I would be surprised if it was less than RMB500 with driver for a day... if you find out more it would be great if you can post your findings back here.
I also know a driver that I have used in the past, he might be available for daily rent so you could try him. I will private message his details to you.