GNU/Linux Ubuntu 22.04 Installing TP-Link TL-WN722N Wifi/Bluetooth Driver Guide
Hi! The linked Tutorial shows you Step-by-Step How to Install TP-Link TL-WN722N V1/V2/V3 Dongle Wireless Driver in Ubuntu 22.04 Jammy LTS GNU/Linux.
And the Ubuntu Jammy TP-Link TL-WN722N V2/V3 Setup is based on the Realtek rtl8188EUS Chipset Driver Installation.
Where the V1 release is equiped of a Qualcomm Atheros AR9271 and so should be Working Out-of-the-Box.
Especially relevant, if you are Unsure about your Wifi Device Chipset then see: How to Get Wireless Adapter Info on GNU/Linux.
GNU/Linux Ubuntu 22.04 TP-Link TL-WN722N Driver Setup Guide
For V2 & V3 see: Realtek rtl8188EUS Linux Driver Installation
Instead for the Legacy V1: It should work Out-of-Box with the bundled ath9k_htc Kernel Module.
Last, in case of Issue with the latest Devices you may see: How to Update Linux Firmware!