First, cut out the main piece as shown on the hat pattern I linked you to.
Second, sew it as shown in the pattern.
Next, draw an arrow. Remember how thick the one was from your tail? You want to make the arrows on the body a consistent thickness. Draw your arrow to be that width.
See the thickness?
Next we work on Appa's face. Remember the brim part of the pattern given? Trace it so that it is one continual piece and then make a slight divot for the lips. Then give it a 1 inch band on top of it so that you can have a place for eyes. I did this on self adhesive stiff felt so that I could adhere the correct shades of grey on it and have firmness without much extra bulk.
Cut out your pieces.
Adhere to the appropriate colors.
Cut out separate nose piece. I used the shiny foot material I bought for the feet and used the other semi suede looking side for the eye band.
Pop holes in the stiff face band you had made in the spots where you had indicated in your drawing. (P.S. this is why I left the lids on the eyes. If you don't, the eyes seem to stare into your soul. Creepy!)
Make sure to hot glue the nose onto the muzzle.
Hot glue the eyes in on the inner side of the face band.
Hot glue them to the eye band.
The hat requires 3 bands. The main one you want to reach from ear to ear and under the chin. You want it loose enough to not irritate, and tight enough that it won't fall off.