I think you're going to need to do some hand sewing to finish up your shrug.
You can follow the instructions in your link, but sew the armscyes by hand rather than on the machine.
Or you could do it this way, which is what I would do (not that I'm one of the super-sewers here):
sew the fashion fabric front and back together at the shoulders.
sew the lining together at the shoulders.
then sew the lining to the fabric around the armscyes, center front, and neckline. The sides and the hem are still unstitched. Turn the whole thing right side out.
Now sew a side seam, fashion fabric to fashion fabric, lining to lining, right sides together, in one long seam. Make sure the intersecting armscye seam matches.
Do the same for the other side.
Now the hem is the only part that's still hanging free. You can do a traditional hand hem. Or you could turn the shrug inside out so that the right sides of the fabric and lining can be stitched across the bottom edge on the machine - but you'll still need to leave an opening big enough to turn the garment right side out again, and that opening will have to be hand stitched.
Maybe someone else knows a better way?