It's a bit of a pain in the arse to get to.
There are a number of different possibilties to do it however:
1. Train to Leipzig Hbf, then airport bus from outside station - check
http://www.thuesac.de/PV/Ryanair.htm for bus times, I think it costs around €12 oneway.
2. Get a lift share at MFG (
http://www.mitfahrgelegenheit.de/ ) - although lifts directly from Dresden to Altenburg are hard to come by. Maybe a liftshare to Leipzig, then another to airport..?
3. Some kind of train/bus/MFG combo - knock yourself out!
The airport only serves Ryanair routes to Stansted and Barca. Facilities are limited but adequate - there's a reasonably priced cafe (which also sells the english papers), an ATM and not much else. After you've cleared security, there's a tiny little convenience store selling monster bags of M&Ms and the like - be sure to get some proper food beforehand. The toilet serves as the unofficial smoking area once in the departure lounge.
Safe journey.