Cruise ship entry

TA, could you, please, tell us what will be your last port of call, immediately before Shanghai? Also, we need your destination immediately after Shanghai. Thank you.

The ship will stop in South Korea Busan a day at sea than shanghai
My flight leaving Shanghai will take me to California than New York

Your itinerary qualifies you for 144 hour visa free transit in entire Shanghai transit area. Quite a larger portion of eastern China. So, you can enjoy your stay in this area, completely visa free for up to 6 days. It is now up to you to decide how many days will you allocate for Shanghai area :)

You should just have in mind to print out your cruise and your flight tickets and also to book at least the first overnight in Shanghai before your arrival there and to have a confirmation of this booking with you.

Bermudian passports are not eligible for 144-hour transit. You need to travel with a British citizen passport (the one WITHOUT "Government of Bermuda" on the cover and says "British citizen" under "Nationality" on the biodata page). If you don't have one then you need to apply for one.