micro DVI on 1st generation macbook air
DVI (DVI = Digital Visual Interface) does never carry sound, it's visual only. Buying a Mini Displayport to HDMI does not help you out either because it's not supported for your Macbook Air.

Here's the list of laptop models that support audio pass-through via MDP

MacBook Air 11" Model 3,1
MacBook Air 11" Model 4,1
MacBook Air 11" Model 5,1

MacBook Air 13" Model 3,2
MacBook Air 13" Model 4,2
MacBook Air 13" Model 5,2

MacBook 13" Model 7,1

MacBook Pro 13" Model 7,1
MacBook Pro 13" Model 8,1
MacBook Pro 13" Model 9,2

MacBook Pro 15" Model 6,2
MacBook Pro 15" Model 8,2
MacBook Pro 15" Model 9,1
MacBook Pro 15" Model 10,1

MacBook Pro 17" Model 6,1
MacBook Pro 17" Model 8,3

and any version later


You can use the Apple MiniDisplayPort to DVI adapter with a cable like this: 
and a separate jack audio cable.