Can I get IOS 13 on my IPAD Air?

No, not possible.

A first generation iPad Air, iPad mini2 and mini3 can only be updated to iOS 12.5.3.

Apple ended major update support for these models of iPad in September 2019 - although they since continue to receive “point” security updates. These devices cannot be updated to iPadOS 13 (or any later major versions of iPadOS) as the internal hardware does not meet the minimum technical requirements (CPU and RAM) for newer versions of iOS/iPadOS.

Some App developers have dropped support for iOS versions preceding iOS 13.x for technical reasons (often because APIs or software libraries upon which they rely have been updated in the new version of iPadOS, making their App incompatible with earlier versions). 

Whilst many App titles remain compatible with your iPad, if you want or need to access to Apps requiring the most recent versions of iOS, you’ll need to consider replacement of your iPad with a newer or current model that supports iPadOS14.

Sorry, I appreciate that this may not be what you were hoping to learn.

I hope this information provides helpful clarity as to your available options - and the limitations of your existing iPad Air.

No.

This will never, ever be possible.

Your iPad Air model cannot upgrade/update beyond any version of iOS 12, currently at iOS 12.5.3.

The 2013 iPad Air 1, iPad Mini 2 and 2014 iPad Mini 3 cannot upgrade beyond any version of iOS 12.

Sorry.

These iPad models have overall internal hardware that is too old, too underpowered and incapable of running all of the new, advanced and internal hardware resource intensive features of iPadOS.

These iPad models will soon be 7 and 8-years old, in 2021.

Sorry.

There is one last thing you can try/attempt, if you need older versions of apps.

If you own another much newer iOS device that can run either iOS 12 and.or iOS/iPadOS 13 and later, you can install the most recent app/s onto that more recent iOS device.

Then go back to your older iPad, open the iOS App Store, go to the Purchases section of the store app, search for the app/s you wish to try and download to that older iPad and see if an older version of the app is allowed to be downloaded/installed to that older iPad.

These newer iOS/iPadOS devices must use your same Apple ID user account and password.

Failing that OR IF NO joy, then there is nothing more to be done.

It's 2021, now.

If you need/want an iPad with an up to date iPadOS with access to current, updated apps, then you really need to seriously consider a purchase of a new iPad model OR a much, MUCH “newer” and less expensive, refurbished or used iPad models, from better known, trusted sources, like other Apple product/device retailers/resellers, locally or online, on the Web/Internet, capable of running the latest iPadOS versions.

There are PLENTY of newer, older iPad models that can run the latest, current versions of iPadOS.

I am sorry that you had to learn all of this, but this does not change any of the facts regarding these very old iOS devices.

Sorry & Best of Luck to You!

The original iPad Air cannot be updated to iOS 13. The iPad Air 2nd, 3rd, and 4th generation can all be updated to the current version of iOS 14.