I looked this pattern up and found it through KnittingPatternCentral. Link to pattern
The designer gives directions about the cast on. She says to use the knitted cast on and actually gives a link to how it is worked here on KnittingHelp.com. Since you are at the beginning of the row you just knit into the first stitch like you normally would but when you have drawn up the stitch loop you put it back on the left hand needle. That adds or casts on 1 st.
You are going to be casting on and then binding off to make the picot border. I looked at what you had said but the stitches didn't add up. When I found the pattern I discovered the problem.
Row 2 actually says:
Row 2: Cast on 1 stitch, bind off 1 stitch, knit 18 sts, (1 st remains on needle), turn.
Now all 20 stitches are accounted for.
You cast on one with the knitted cast on, you now have 21 stitches on the needle in your left hand. Knit 1, knit another 1 and then lift the first over the second and off the end of the right hand needle (in other words bind off one). You will now have 19 stitches on the left hand needle. Knit 18 of them, leaving one unworked, turn like you were at the end of the row and be prepared to work in the other direction.
You are working what are called short rows because you turn to knit back in the other direction short of the end of the row. This is how you are going to shape the wedge shapes for the swirl.
You will learn a lot making this cloth. Have fun.