How can goat milk be prepared to reduce the 'hairy' after taste it has?
I think I understand what you mean about "hairy" taste; I think of it as a "goaty" taste. Unfortunately, the diet of the goat contributes much more to the flavor of the milk, than sheep or cow diets/milk. This is one of the reasons that goat milk is a favorite when it comes to cheese making. It adds a little funkiness to the cheese.

Now all that said, how are you drinking your milk? I love goat yogurt and cheese, but I'm not crazy about goat milk in my coffee. Part of this has to do with the temperature at which your consuming the milk. I usually eat my yogurt cold, and the goatiness is noticeable, but not as much as when I try to drink it in my coffee. Warm/hot foods tend to "taste" stronger, while cold foods kind of numb your taste buds. 

I know that I haven't answered your question. I think if you drank it for a while (say two weeks), you'd notice it less and less. Kind of like moving from whole milk to skim milk.