I can’t get my pencil to pop up when plugged into my iPad air

As already clarified, you will not see a connection notification graphic as you do with 2018/2020 models of iPad Pro.

As described, the issue that you describe is often attributable to a failing or failed Pencil battery - that will not accept or hold any charge.

Assuming that you do not have any other iPad charging issues - and that the Lightning Port of your iPad is free from debris - whilst the Pencil is plugged-in to the Lightning Port of your iPad, verify that the Pencil is recognised:

Settings > General > About

Scroll to the bottom of the page - and you should find two identical entries for the Pencil (one each for the physical and Bluetooth connections).

The tiny internal Li-ion battery is susceptible to permanent/irreversible damage through being left discharged for long periods. Even some “new” pencils can exhibit signs of failure out-of-the-box if they are “old stock”.

It is essential that if you have an Apple Pencil that you charge it regularly - whether used or not - so as to protect the battery from deep-discharge. Do not allow a pencil to remain in low-charge state for any period of time - as the internal battery will fail, rendering the Pencil useless.

If the Pencil Battery has failed, the only remedy is to replace the Pencil. If purchased directly from Apple, if the battery has failed and your pencil is within its one-year warranty, you should look to having it replaced when the Apple Stores eventually reopen.

If the Pencil battery is not the cause, here is one of the better troubleshooting guides for Apple Pencil issues:

https://appletoolbox.com/apple-pencil-not-working-heres-our-troubleshooting-guide/

I hope this information proves to be helpful in diagnosing and potentially resolving the problem.

Did you turn on Bluetooth? And did you charge your Apple Pencil? Those are usually the most common mistakes that people do when connecting their Apple Pencil. If that’s not the case, restart your iPad, go into the Settings app and click on Bluetooth. Make sure the Bluetooth toggle is turned on so that it can read your Bluetooth devices such as your Apple Pencil. On the same screen, look under My Devices. If you see your Apple Pencil show up, tap on it to connect. A firmware update on the iPad may help as well.

Can you explain exactly what you anticipate seeing when you connect your first generation Apple Pencil to your iPad?

Unlike the second generation Pencil with 2018/2020 iPads Pro, connection of a first generation Pencil to a compatible iPad (with a Lightning connector) results in no obvious/overt indication of connection to the iPad.

Lotus is correct. Go to Settings > Bluetooth

Do you see the Pencil there and does it say connected?

I kept a copy of this as some people have had issues with their Apple Pencil connecting to their iPad.

Perhaps the solution in this link may help --> https://discussions.apple.com/thread/251382715

Being an iPad Air 3 (suggested by the device tags - which I had initially missed), if the iPad is running iPadOS 13.5.1 (as yet unconfirmed), this device-specific issue is a very likely candidate.

The described workaround will probably fix the problem.

Is this something which happened recently or has the pencil been uncharged for any length of time recently?

I hope it does.