Can I patch up a straw hat?
Your hat has a very open structure, so, to fix a hole invisibly is hard/impossible, as you need to weave in material where it's missing, but then you got a double amount of material at the edge of the hole.
However, since you 'just' want to make it look nice, you could


reweave the hat (with either straw, wool, or any other material of thead) about as thick as the original material. It's obviously a lot of work and you loose the open structure. But you don't have to reweave the complete hat, you could skip the rim, or any other parts without holes.
Stitch/embroider a ribbon along the edge of the crown, using whip stitch (If you don't know this stitch, see https://nl.ifixit.com/Guide/How+to+Sew+a+Whip+Stitch/21025) Make the stitches large, and be sure to cover the hole with the ribbon.
Stick a flower in the hole, and attach it in a creative way. :-)