144 visa with transit in beijing and shanghai

Unfortunately, your itinerary is not eligible for any kind of visa free transit scheme. Simply speaking, when traveling on 144 hour visa free transit, you are not allowed to have two mainland cities (landings) in a row regardless of the nature of those landings. I understand that your main problem is the fact that there is no nonstop flight from Ulaanbaatar to Shanghai, but you will have to adapt your itinerary by taking your flight from Ulaanbaatar to Seoul and then to Shanghai and then from Shanghai to Bangkok...