By 1936, Shanghai had been occupied by Japan which invaded the city in 1932. Though Japan hadn't officially declared war until 1937 (the start of the Second Sino-Japanese War), they had been taking up chunks of China starting with Manchuria in 1931. Even though Japan occupied the city, the French and British Concessions and other Western imperial powers in the city declared neutrality, so they were left alone. (The Western powers gained most of those chunks of the city during the First Opium War in 1842 in order to have good access to trade, mainly opium, in China.) Many Chinese in the Old Town/Chinese section of the city fled to the foreign concessions to avoid the Japanese.

Japan had a really good incentive for taking Shanghai as it had become one of China's biggest trading centers, if not the biggest (Hong Kong might top that though). It's location relative to Japan had also benefitted Japan's own trading relations with China. Opium was a prominent product being traded freely throughout the empire since it was forced to be made legal during the Second Opium War. I haven't done research on Shanghai specifically, but especially early on during Japan's invasion, there were a lot of Asian elites that were eager to be a part of the Japanese Empire, as it was the only non-Western country that had modernized to the point of being an equal member on the world stage with the Western powers. Japan also promoted this through its propaganda as the "Greater East Asian Co-prosperity Sphere" which would uplift the rest of Asia to modernity, with Japan as the leader of course. A lot of their invasion into China was ideologically argued as an attempt to expel Westerners from Asia. So while the Western powers in Shanghai had declared neutrality, I can't imagine that Japan was too happy with them there. They of course took care of that later when they allied with Nazi Germany and began WWII in 1942.

There's a really great Chinese tv show about Shanghai in 1939 called The Disguiser, or Weizhuangzhe. You can find it on Viki.com with English subtitles. In case you can't watch it all in time for your game, here are some of the themes that I think you might find most important. The show covers the interesting dynamics between the Nationalists and the Communists during this time period. (I say interesting since after WWII, they almost immediately descend into civil war.) During the war, they worked together, grudgingly, against the Japanese as Chinese people, but ultimately the Nationalists are depicted as morally corrupt. The main character, who is a spy for the Chinese Nationalists, eventually converts/changes sides to the Communists at the end. It also touches on the issue of collaboration of the Chinese people in the puppet government, which is a big topic in general in Asian WWII history. The main character is a spy, but his brother works for the Japanese-led Shanghai puppet-government. The brother colludes with the puppet government, but this is only ok in the eyes of the show because he is also secretly a spy. Though, he bemoans the fact that he will never be able to show his true colors to his countrymen as a true Chinese. The brother's girlfriend however, who is also Chinese and works for the puppet government, is portrayed as evil and inhuman and doesn't deserve sympathy and ultimately dies at the hands of the brother.

EDIT: So after a little more digging, I realized I got some dates wrong. Apparently, the Japanese didn't fully occupy the city until 1937 after the Battle of Shanghai, when the Nationalists tried to take it back. Before that, the Japanese just had some troops stationed around the region. Most of my response still holds though, the timeline was just a bit off.