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That's my edit of the horns. I couldn't redo the whole hat because well, study time! Nor did I have time to AA.
Anyways my point is don't do 1 pixel then 3 pixels and then back to 1 pixel, it looks bad.
Use the 1-2-3 rule it makes the art look more round and smooth. You can check on the right side that I did an example.
I did edit other pieces around the helm, if you could see the bottom where the chin would be, yours is sharp edges and jaggy, while I edited that part and it looks smoother. Try to follow my example. As for the horns, xD Yeah they're large but they help make it look like a more-realistic viking hat. Have you seen google images of viking hats? The horns are normally large.
For shading, you don't need to use so many colours, try not to use so many and work on finding colours that contrast with each other to make your hat look like it wants to pop out.
Sorry I couldn't do much, hope my example will help some of you out.