I do not have time to replace my passport. Will I be ok to travel and transit through Shanghai?

Fiona, how long will you stay in Shanghai? If the time is more than 72 hours, you have to apply for a transit or touist entry type. In this case, the reamaining validity of your passport should be more than six months. I'm afraid the consular office won't approve of your application. If your layover is not more than 72 hours, perhaps you can make use of the 72 hour free transit policy. In this case, your passport may work. However, I still suggest you confirm this with the embassy.