I looked at the PDF of the pattern on Ravelry
and was a little surprised to see that the designer recommends only a 24" circular for 130 sts in bulky-weight yarn. Wow. Those are going to be some very crowded stitches!
She also gives a gauge of 3 sts and 5 rows = 1 inch in seed stitch (the alternating knit/purl pattern she's given in the instructions). If you're working this on straight needles, it *might* work to do it "sideways"; in other words, to case on for the width rather than the length.
The instructions say to knit seed stitch for 15 inches. If you crowd the straights a little bit (are they 14" long?), a 15" width will be OK. At 3 sts = 1 inch, you'll need 45 sts for 15 inches. I think most people find it easier to work seed stitch over an even number of stitches, so either 44 or 46 will work (maybe 44, since that's a little less crowding the needles).
Then go for 131 rows (the original number of stitches that were to be cast on), or as long as you want the cowl to be. You can then join the two ends in any of several manners and have a nice, long, full cowl.
HTH, because where can you purchase circs this late on New Year's Eve? Or do you have a friend who can lend them to you for a few weeks? (I'd ask for 32" at least, though, if not longer....)