My iPad Air does not give an option to update software to 13.5.1. Instead it says that software is up to date at 12.4.8. How do i update to 13.5.1?

A first generation iPad Air can only be updated to iOS 12.4.8. If instead you have an iPad Air2, or the current Air3, these do support the current version - iPadOS 13.6.

Apple ended update support for iPad Air1 in September 2019. An iPad Air1 cannot be updated to iPadOS 13 (or any later major versions of iPadOS) as the internal hardware does not meet the minimum technical requirements for new 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 Air1, 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 iPadOS13.x.

If you do have an Air2 or Air3, it should be possible to simply update your iPad, without a PC/Mac, from Settings:

Settings > General > Software Update

Sorry, I appreciate that this may not be what you were hoping to learn if you do, indeed, possess an iPad Air1.

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

Your iPad is not lying. It can't support iOS versions beyond 12.4.8, because of hardware incompatibility, and there's nothing you can do, sorry.

Regards

Giulio