how can i update letest verson of ios in my old ipad air (3)

Assuming your iPad AIR is a 2nd (or higher) generation model which supports iPadOS 15 (Identify your iPad model - Apple Support), and being aware that to perform an update procedure wirelessly, you need to connect your device to a reliable and fast Wifi Network with internet access (no cellular data, unless you use a 5G network, no personal hotspot on your device, no VPN connections are admitted). AND I suggest that you keep the device plugged into power during the whole update process.

If you don't have a wifi network, "borrow" one from your friends/neighbors, or ask for help at an Apple Store or Authorized Service Provider or Authorized Reseller.

That said, see this suppport article:--> If your iPhone, iPad or iPod touch won’t update

If the issue persists, try to update it via computer:

Learn how to update your device manually if you're using a Mac with macOS Catalina or later, or if you're using a Mac with macOS Mojave or earlier or a Windows PC.

If you still get no help, contact Apple Support directly, using this link:--> Get Support

Regards

Giulio

Whether or not you are able to update your iPad to a newer version of iOS/iPadOS will be dictated by your iPad model - and the currently installed version of iOS. You can find this information here:

Settings > General > About

Not sure which iPad you have? Identify your iPad model.

Here is a list of the highest version of iOS supported by the older models of iPad (for simplicity - “generations” are shortened to a number) that are restricted in their ability to be updated:

iPad1 - iOS 5.1.1

iPad2, iPad3 and iPad mini1 - iOS 9.3.5 (WiFi Only models), iOS 9.3.6 (WiFi & Cellular)

iPad4 - iOS 10.3.3 (WiFi Only), iOS 10.3.4 (WiFi & Cellular)

iPad Air1, iPad mini2 and iPad mini3 - iOS 12.5.5

Newer models, not appearing in this list, are all able to be updated to the current version, iPadOS 15.0

If your iPad is capable of being updated to a more recent version of iOS/iPadOS than is currently installed, these support pages outline both the available methods by which the update can be performed - and troubleshooting should you experience difficulties:

Update your iPhone, iPad or iPod touch – Apple Support

If you can't update or restore your iPhone, iPad or iPod touch – Apple Support

If your iPhone, iPad or iPod touch won’t update - Apple Support

iOS update and restore errors – Apple Support

Be aware that your iPad may be too far behind the current version to be updated directly from iPad settings. As such, you will likely need to use a computer to perform the update. You will need access to either a PC (with iTunes installed) or a Mac (with iTunes or Finder - as appropriate for the installed version of MacOS). Ensure that your installed version of iTunes has been updated to the most recent version; the essential device drivers are included.

Download and use iTunes for Windows – Apple Support

Update to the latest version of iTunes – Apple Support

Update software on iPhone, iPad or iPod touch in iTunes on PC - Apple Support

You should note that for your iPad to install an available update, its battery must be charged to at least 50% capacity - and/or be connected to its Power Adapter. Similarly, the iPad can only be updated over a standard WiFi connection - or be updated by a PC or Mac using iTunes or Finder - over a USB connection.

A little hard to help/advise you when you provide little information about your iPad Air.

There are 3 models of older iPad Air models.

Which year, model and screen size do you have?

Can you, please, confirm what year and model iPad do you have/own?

The “A” model number can be found on the aluminium back of your iPad, in very small print, right under the larger bold print “iPad” on the back of the iPad.

What iOS/iPadOS version is this iPad, currently, running?

What is the rated, internal data storage space/capacity of your iPad model and how much of this free, internal data storage capacity is left inside of your iPad model?

All of that additional info can be found in the Settings app, General settings panel, About.

Please try to attempt to answer all questions asked.

Thank You