How can I recover my lost AirPods case?

If these are AirPod Pro 2 or newer, you can locate your case, even if the AirPods are not in it:

...

With AirPods 4 (ANC) or AirPods Pro 2 and later, you can also separately mark each of your AirPods and the case as lost, in the event that you lose just one or your AirPods are separated from the case.

...

copied from: Find your lost AirPods with Find My - Apple Support

In any other case you might have to buy a new case:

Get AirPods case replaced by Apple

I have AirPods Pro 2nd gen and just tried to locate the case.

If my AirPods are not in the case and I open Find My -> Devices, the app recognises this and shows the AirPods and the case separately. And, if I tap on the case, I could use "Play Sound", or "Find" to locate the case.

Did you try to use the option to get notified or leave your information?

Are your AirPods connected to your iPhone? If yes, the app should be able to locate them and also see, that your case is not near you.