Hey there, RedstoneRosh12!
Thanks for reaching out in Apple Support Communities! From your post, we can see you've been using Switch your AirPods to another device, and have found that you've been having trouble getting the AirPods Pro to sync.
You're using a great article! We'd also like you to try this guide: If your AirPods won't connect. Although you're unable to update the iPad Air, you'd want to make sure to update your iPhone 13 Pro to the latest iOS version and follow the checklist under "If you can't connect to your iPhone, iPad, or iPod touch".
To answer your compatibility question, here's the tech specs: AirPods Pro with Wireless Charging Case - Technical Specifications. If you're using the first generation of iPad Air, it would not compatible as listed in this guide.
Take care!