US --> Barcelona --> Shanghai (5 days) --> Tokyo

Your itinerary, under the condition that you have mentioned all your stops is definitely eligible for 144 hour visa free transit in Shanghai area.

You will print out your flight tickets and your accommodation booking confirmation and have those printouts with you. Also, you will declare in Barcelona that you will travel to Tokyo via Shanghai using 144 hour visa free transit. After a short inspection of your flight tickets, you will be allowed to board the plane.

Upon landing in Shanghai, you will approach 144 hour visa free transit where you will fill out a blue form and shortly after you will be get a sticker in your passport (everything free of charge) and you will be allowed to exit the airport.

When you exit the airport, you are allowed to travel and stay within the area comprising of:

Shanghai City
Jiangsu province
Zhejiang province

Quite a large area and Hangzhou is within this area, so you are allowed to travel and to stay there as you wish.