You can get some of the cash shop hair cuts with the free coupon, but it's random. My friend and I checked it up ( after we both got buzzcuts on it ftl) and I'm pretty sure there's like 8 different cuts you can get.
Well,there is no way to change it without points or nx. Every possible haircut with that coupon is ugly, you will never get a good one. That was the whole point on Nexon's side,they want you to buy nx.
These are the Amoria haircuts: http://www.maplefantasia.com/gms-male-hairstyles-amoria/ But note that the quest doesn't give those haircuts, they give you a random one (male) from this list: http://www.maplefantasia.com/beauty-or-beast-male/
I think the quest is not worth doing. There are two hairstyles (in my opinion) that are not hideous (but they're not exactly great looking cuts either). One of them is a haircut you could've picked when you made your character anyways. So there's really only 1 new cut that's decent, the rest are all horrible.
So 7 ugly cuts, 1 decent cut, and 1 default cut.
Counting on Nexon to finish the patching early is like hoping Sally will get a girl and stop getting off of trolling GameFAQs.