Arrival/departure card

When the 144 visa free transit rules speak about the regions, it is not wider geographical regions, but transit regions within mainland China. So, it is not possible to travel to two mainland Chinese regions in a row, but traveling within one geographical Asian region is perfectly eligible. The basic rule of all the rules for 144 hour visa free transit is that the immediate points before and after your mainland Chinese destination have to be in two different countries or eligible regions. In other words, your itinerary:

Singapore - Beijing - Japan

Is definitely eligible for 144 hour visa free transit in wider Beijing transit area that encompasses Beijing and Tianjin cities as well as the entire Hebei province. It is important that your Singapore to Beijing and Beijing to Japan flights are of nonstop type.

As far as arrival/departure cards are concerned, you will get one aboard the plane from Singapore or you will obtain one upon landing in Beijing. This is not a problem at all.