This looks like a fun project to work. Basically, you're working short rows on the even rows. Just turn the work as you would if you were at the end of the row. The directions tell you how many sts to leave unworked on the left hand needle for each of these rows. Don't worry about them; you'll get to them later. In the end you bind off all sts and leave about a 10" tail to gather the bound off sts and sew the cloth up.
You can certainly use a circular to make this. Knit back and forth on it without joining to knit in the round.