Can I completely shut down my Apple MacBook Air to save battery life?

Yes, if you shut down the computer it should not drain the battery.

To properly shut down the Mac, choose Shut Down... from the Apple menu  at the top left.

Simply closing the lid of the laptop does not shut it down, it simply idles it, which will still draw down the battery.

You can tell when the computer is shut down when the keyboard lights go out and the screen shows no hint of light. If you engage the caps lock key before you shut down you can watch as the key light turns off indicating the computer is now off.

It's also a good idea to leave it plugged in when not using it. MacBooks are battery-capable devices but leaving it plugged in when not using on battery is recommended.

Ok thank you for your support and going to give me try the new apple MacBook M3 right?