there used to be a bus went between the shanghai pudong airport (the one used by most international flights) and nanjing. but it ceased last year because it was not very profitable.
there are many buses running between shanghai and nanjing during the day. almost one every half hour. this one however goes to another city. the traffic in and out of shanghai each day from its surrounding area is huge. you have to really listen to catch people speaking shanghai dialect on the street.