The building looked familiar, and I identified it as Dupleix Apartments (built in 1918) on 233 Anfu Lu. The author's family lived on the second floor for two years from early 1927 till 1929. Richardson says "this was the first apartment house built in Shanghai."
Notice how much the building has sunk! The base of the column at the entrance is underground now. Presently the lobby is at the same level with the ground.
It is sad those exquisite lanterns on the sides of the entrance are gone, as well as the fanlight window with a spiderweb design. Instead we now have a standard ugly-green grill door and air-con units on the windows. The shack on the left is part of a long row of added houses clinging to the side. I much preferred the natural-looking dappled brick surface (you can see it on my photos below) that got replaced with monochrome pink a few years ago.
Before the "restoration" I visited an apartment for rent on the first floor. This was not necessarily Richardson's childhood home because there are two apartments on each floor. The layout is elongated: a long corridor runs all the way through, with rooms on one side. In the far end there is a dining room that has the best light and the best view of a chaotically overgrown garden that I've seen in Shanghai. More photos from that visit are in my Flickr set.
These are more details from the book about (high) life in Kingsbury Apartments:
(I wonder where is that park now?) Update: found the park.
(There is no such rounded staircase in Kingsbury/Dupleix. This looks like another building on the same block: 255 Anfu Lu).