Do I qualify for the 144 hour Free Transit China Visa?

Your itinerary:

Japan - Shanghai - US

Is definitely eligible for 144 hour visa free transit and you don't need any kind of visa. All you have to bring with you is your valid passport and your flight ticket from Shanghai to US. You should just print out your flight ticket and have it with you and please, just have in mind that this printout shows that long 14 digit number of your flight ticket.

You will be probably asked in Japan about your intentions and you will tell them that you will travel to US via Shanghai using 144 hour visa free transit. You will present them your flight ticket as a proof.

Upon your arrival in Shanghai, but maybe already on board of the ship, you will get a blue (beside a yellow form that is an arrival/departure card) form that you will fill out. You will hand this, along with your flight ticket to the Chinese authorities. After a few moments, you will get a sticker in your passport for free and be allowed to exit the cruise terminal. Basically, this is it.

You should not apply in advance, but you will just follow the above mentioned, simple instructions. Also, please, just use correct terminology because it is not a visa, but 144 hour visa free transit or in other words a transit without a visa.