Your translation of the text is not correct. You have to read it as two different remarks:
- Erwerbstätigkeit gestattet
- Kartennutzung bis 16.07.2020
This translates as:
- Employment allowed
- Card usage until July 16th, 2020
The right to work is not limited in time, but the usage of the card. The reason is as you probably have guessed that the residence permit card is only valid together with your passport. Your old passport number will also very likely be mentioned on the residence permit. If you get a new passport, you will also have to get a new residence permit card (eAT).
Again: Be aware that your residence or work permit does not expire on July 16th, 2020, it is just the validity of the eAT card. German authorities are very well aware of the practical impact of the corona restrictions, so I wouldn't worry at all if you are just planning to stay in Germany. You might however run into problems if you plan to travel abroad, especially leave and/or reenter the Schengen area with the expired card.
How to proceed depends on where you live. In Germany, the foreigners authority is a municipal office and each office seem to deal differently with all the corona related problems. Unless your local office has a web page telling you how to proceed, your only option is to try to get in touch with them and ask. Some municipalities now allow 'simpler' things like in this case the renewal of an eAT card (as long as the residence right itself is not expiring) to be done by e-mail, fax or mail. In Frankfurt, you could for example simply mail the application with a scanned copy of your new passport to the authority and you would get the new eAT card sent by mail to your home address.