I'm using the mesh found in this link which is animated using shape keys. I'm wondering how I can do the same animation but using bones (armature) instead? So if anyone could please advise.
You can drive those shapekeys with bones, this is actually the preferred way of doing facial animations. This video explains in a detail how to setup the drivers and hook the bones to the shapekeys. In the first part he creates all the face-expression shapekeys and in this second one he drives them with bones.
I've been experimenting with facial armatures! I use envelope weights, weight paint, and bone constraints to mirror sides of the face, and I was trying to make a weird sort of pentagraph-like thing for shaping the outer mouth and cheeks
you could use the shape driver constraint add-on to control those shape keys. The bones could be constrained as the drivers. Here's the link to the website and video guides you through how to use the add-on: https://www.blendernation.com/2015/05/18/addon-shape-driver-constraint/