72 hours Visa free entry in Shanghai

Petr, here we have two separate cases.

Your wife as New Zealand citizen can enter China visa free for up to 15 days. She does not need any visa free transit or similar stuff. She will just fill out an arrival card that will be dispensed on board the plan that you will arrive in Shanghai and she will tick transit box. She will then just proceed to the immigration with her passport and with the filled out arrival card.

You as Czech citizen, you will not fill out an arrival card. Your first step upon arrival will be to approach 24/144 hour visa free transit counter (your wife can make you a company, of course) where you will apply for a temporary entry permit to be allowed to exit the airport. Here you will present your flight tickets. This application is free of any charge and after a short check, you will be granted a permit. The fact that you will tell that your wife will exit the airport will increase your chances to be allowed to exit with her.

After you have received a permit, you will both head to the immigration in order to exit the airport and to enjoy some sightseeing after that.

Please, have in mind that you will be traveling here on 24 hour visa free transit because your layover is shorter than 24 hours. Enjoy your journey.