What is its goal?
The main goal of the Brenizer method is to mimic the shallower depth of field you get with a medium or large format camera without using a medium or large format camera.
While larger formats do not, inherently, give you shallower DoF, the fact that larger formats yield a wider field of view tends, if you try to compose an image identically as you would with a smaller format, cause you to either get closer to your subject and/or use a longer lens, and both of those changes can reduce the depth of field.
You typically shoot this type of panorama with a telephoto or short telephoto fast portrait lens (say, a 135/2, or 70-200/2.8 on full frame, or an 85/1.8 on crop), with the aperture set near to wide-open to get the shallowest DoF possible for each member image.
Is it a merely post-processing job?
Yes, and no. The panorama stitching, certainly is a post-processing job. But given that some frames may be entirely out of focus, this also requires a bit of planning and setup on the shooting as well. You may want to use a specialized panorama head, not so much for rotating around a no-parallax point of the lens, but more to track the coverage of the scene, given the narrow angle of view that longer lenses yield.
And you certainly want to have exact focus, locked in with manual focus--and you have to have subjects that can hold still while you shoot ALL the member images. You also need to be experienced enough with panorama shooting to make sure you have enough coverage for the scene you want with no gaps, and enough coverage through time to be able to edit out ghosts or clones that can easily by caused by anything moving through the background.
Can it be applied with any lens?
It can, but the effect is unlikely to be realized without using a long/fast lenses, since wide and slow lenses tend not to exhibit out of focus blur as easily unless used at very close subject distances, and the whole point here is to get a shallower depth of field at normal or slightly farther subject distances.
Is there a preferred scenario?
This is most often used for portrait photography. Typically it's done outside with a picturesque view surrounding the subjects, and the shallow depth of field is used to emphasize the couple/subjects against the backdrop. But any time you wish you could have shallow depth of field, but your lenses, sensor format, and working distance prohibit it will pretty much work for this--as long as you have a lens and aperture combination that can achieve background blur at the distances you want to work.