Three months is enough. Many riders do it in less time. Few take longer.
The ACA 2012 Touring Bike Buyer's Guide says, "Factory bicycles suited for fully-loaded touring can be had for roughly $1,000 to $2,000, with very few offered below the $1,000 mark. If you can’t afford that, a light touring bike might be had for $500."http://www.adventurecycling.org/default/assets/File/AdventureCyclist/OnlineOnlyFeatures/BuyersGuide/201204_BuyersGuide_Wider.pdf
Below $1000, the guide includes:
Motobecane Gran Tourismo ($800)
Raleigh Bicycles Port Townsend ($850)
Novara Safari ($899)
Marioni Tourismo ($875/$1000); Tourismo Extreme ($925/$1050)
Keep in mind that you probably need to spend more on gear. From your AT hike, you probably already have most of the non-biking clothes and camping gear you need, but you probably need racks and panniers, some spare parts and tools and some biking-specific clothes, gloves, shoes and helmet. You'll also need to save some money for food and camping fees en route, and for transportation to the start and home from the finish.
Most bicycle tourists use upright bikes. Those that use recumbents, however, are passionate about them.