Software update

Are you certain that you have an iPad Air - and not an iPad4? If you do indeed have a first generation iPad Air, you should receive indication of iOS 12.4.8 being available.

If your iPad is currently running, or cannot be updated beyond iOS 10.3.3/4 - then the behaviour you observe actually suggests that you have an iPad4. This can be confirmed from your model number:

Settings > General > About

Update support for iPad4 ended September 2017. The final iOS releases for this model were iOS 10.3.3 (WiFi Only models) and iOS 10.3.4 (WiFi & Cellular). iPad4 cannot be updated to any later major version of iOS as it lacks the minimum hardware requirements needed to run more recent versions.

Many App Developers have also dropped support for older devices and iOS versions preceding iOS11/12/13 - and have withdrawn older versions of their Apps from the Apple App Store. Withdrawal of Apps supporting older models of iPad is making their utility and usefulness significantly more limited. Once compatible versions Apps are withdrawn, you’ll not be able (re)download or (re)install them.

If Apps that you need to use now have higher minimum iOS-version requirements, your only option is to replace your iPad with newer model. Even the basic current iPad7 supports current iPadOS 13.x - and has hardware specifications that will vastly outperform your old iPad.

I hope this information is helpful - if only in clearly identifying the likely model of your iPad, its limitations and in outlining your current options.

The original iPad Air, released by Apple in 2013, can use iOS 12.x.

Perhaps you are using an antiquated version of iTunes and/or your computer’s operating system to attempt the update.