As the Asheville resident I will steer you to the promised land...
The South Slope (Right Downtown) has 7 breweries in a 2 block radius each around 2-3 minutes apart. Start at Catawba and then check out Twin Leaf, Wicked Weed's Funkatorium, Green Man, and Burial Beer Co.
If you have time, head to Mills River to check out the new Sierra Nevada brewery. Its mind blowing and they have an incredible beer selection in the Tap Room.
Are you staying in downtown?