How to reinstall macOS X Mountain Lion in MacBook Air?

If your Mac starts up to a question mark

A folder with a flashing question mark means that your startup disk is no longer available or doesn't contain a working Mac operating system.

What he should have done is this: What to do before you sell, give away or trade in your Mac

You can recover the Mac using the steps in the first linked document: If your Mac starts up to a question mark

If the question mark persists, follow these steps:

1. Press and hold the power button on your Mac for up to 10 seconds until your Mac turns off.
2. Turn on your Mac and press and hold Command (⌘)-R immediately to start up from macOS Recovery.
3. While in macOS Recovery, use Disk Utility to repair your startup disk.
4. If Disk Utility hasn't found any errors or has repaired all of the errors, reinstall macOS.
5. If you still need help, please contact Apple Support.

Edit: Snap! I see P. Phillips has posted just ahead of me. Two for the price of one.

If your Mac starts up to a question mark

A folder with a flashing question mark means that your startup disk is no longer available or doesn't contain a working Mac operating system.

If the question mark appears only momentarily before your Mac starts up normally:

1. Reset NVRAM.
2. Make sure that your preferred startup disk is selected in Startup Disk preferences.

If the question mark persists, follow these steps:

1. Press and hold the power button on your Mac for up to 10 seconds, until your Mac turns off.
2. Turn on your Mac and immediately press and hold Command (⌘) and R to start up from macOS Recovery.
3. While in macOS Recovery, use Disk Utility to repair your startup disk.
4. If Disk Utility found no errors or repaired all errors, reinstall macOS.
5. If you still need help, please contact Apple Support.

If you have access to another Mac which is compatible with an OS this MBAir supports, then you can use the other Mac to create a bootable macOS USB installer. Here is an article which lists which Apple hardware are compatible with which versions of macOS:

https://eshop.macsales.com/guides/Mac_OS_X_Compatibility

If a Mac can boot to the desktop, then click on the Apple menu and select "About This Mac", otherwise enter its serial number here in order to get the exact model of the Mac:

Check Your Service and Support Coverage - Apple Support