Carcassonne on a Monday?

Ive never heard of a city being closed. Some shops may be closed for part or all day on Mondays. Unless you're looking at the day after Easter The Castle will have its usual opening hours.

touristy towns like this tend to shut down on Mondays -- I don't have experience of Carcassonne particularly but have visited other medieval walled towns that are heavily touristed and had trouble so much as finding a cup of coffee on a Monday. Basically most shops, restaurants, museums etc were closed. It is often the weekly day off just as is true of markets and market streets in Paris and other cities. They stay open Saturday for tourists and then close up Sunday afternoons and Mondays.

I would email the Carcassonne tourist office and ask them.

What do you mean? La Cité is the restored walled city. It doesn't close. French children have two-week vacations around Easter, so you can be sure the historical area will be buzzing. In the other part of Carcassonne, restaurants and cafés that close on the weekends are open during the week. Not everything shuts down all day Monday.
Can you explain what you mean by the city will be closed.

No one really cares about the modern parts of these old cities. One strategy we use is to avoid touristy quaint towns on Mondays having learned first hand that this is the worst day to visit them. Another strategy if you want to go Monday is to identify several restaurants and shops in the old town and google their websites and see if they are open.

If it is Monday April 10, 2023, it is Easter Monday and a public holiday in France.

This may be why you have been informed of possible closures

This will mainly be shops and supermarkets. But the city will not be closed!! Unless she reconnects with Catharism and is besieged by crusaders or the Visigoths (which I imagine you are not)