Billboarding on a single axis

I think the title roughly describes what I want to do.

As I understand, billboarding allows you to make an object always face the camera.

But what I’m trying to do is have my ‘billboard’ stick up from the model and have the X and Y axes constrained to normal perspective.

That probably makes no sense whatsoever, so have a picture instead: (apparently I’m not allowed to post images or links)

This example shows a rectangle on a plane with one side marked by a red ruler. This ruler will always stick up from the model so it’s always visible, whether viewing top-down, or side-on.

But I don’t know how I should go about trying to do this.