I purchased a new Apple mouse. The cable does not plug into my MacBook Air. 13" 2014. What is my solution to this?

Apple only sells wireless mice these days. Where did you buy it, and can you exchange it for a more recent or current production one?

https://www.apple.com/shop/product/MK2E3AM/A/magic-mouse

If the one you bought does not have the appropriate connector then you will need to find an adapter, which may or may not work. It's not an attractive solution. You can also charge the current production Magic Mouse with a USB-C power adapter.

When it charges will it work? Currently shows up in the Bluetooth window as connected and charging.

Question answered. Yes it works. So the only issue is the charging and that can be done with the USB-C power adapter.

👍

Yes. You can use any charging cable and adapter that fits.