Yesm you could leave from the airport to downtown Shanghai. Have you booked a hotel at the airport? Actually, it is suggested that you book a hotel in downtown Shanghai at Jing'an Temple area, Nanjing East Road or Lujiazui Area.

Here are two suggestions:
1. you could take the airport shuttle bus Line 2 to Jing'an Temple. From here, you could walk along the Nanjing West Road at night until to the Bund. On the next morning, you could take the bus back to the airport as well.
2. you could take the Maglve train from Pudong Airport to Longyang Road station. Then, please transfer to the subway Line 2 to get to Nanjing East Road or Lujiazui. At night, you could visit the Bund and its nearby area.