My Air charger stop to respond. Can I use the USB-C charger with this adapter to charge the Air?
That will not work on MBA 2013,
It has only one port where you can charge it (the magsafe plug).
You will need new power charger for your MB Air 2013.
till the USB/C came out the USB were only data ports (with 5 Volt) and could not handle the power needed to charge a Mac.
To charge your 2013 MacBook Air you need 16-18 Volts input and correct 5 pins plug.
No, it's not possible to do that. The adapter you have pictured is the Nonda USB-C to USB-A adapter. It is commonly used for connecting older USB-A hardware (such as USB pen drives) to newer USB-C computers.
One of the big differences between USB-A and USB-C is that USB-A does not allow for transferring enough power to charge a laptop (i.e. it is limited to only few watts compared to USB-C). In addition USB-A is used to charge devices from the computer - not the other way around.
I have seen multiple adapters for using old MagSafe chargers with new USB-C computers. Unfortunately, I haven't seen any for charging old MagSafe 2 computers with a USB-C charger.