I like this neckband too and it works well iwith worsted weight yarn. You can continue the neckband as you suggest, in the rib pattern and then fold it over, bind off and use a long tail of yarn to stitch the band to the inside.
If you don't like the rounded edge that this gives, you can purl a row across the ribbing which will form a fold and then continue with the neckband. This also works if you find the ribbed facing too bulky. After the purl row, continue in stockinette for the facing. It's not quite as stretchy but it works.