How to upgrade to latest OS on Macbook Air?
The new macOS is not displaying because you are not signed into the App Store with your AppleID.

Since you are in the category of "OS X 10.10 Yosemite or earlier" (10.9 is before 10.10), you might have some trouble signing in because your OS does not support two-factor authentication.

Use two step verification:

When you open the App Store and try to sign in, you may see some text in red saying "Please type your password followed by the security code".

Check your phone. Your phone may ask if you allow the login from a certain device; click "allow". It will then show you a pin number.

So if your password is "hushpuppies" and the pin on your phone is "123456", type "hushpuppies123456" in the password field.

After you have signed in, you still might not see anything in the "updates" tab. Try looking at the "featured" tab to see if the OS is there. Once you find it, (in my case, there was a banner in the featured tab for "Catalina") click the "get" button and the download will start.

Run the downloaded installer and follow instructions.