System restore macbook air with broken screen?
You should be able to connect it to an external monitor.  All you need to do (in principle) is this: 


Close lid, let computer go to sleep
Connect both the power supply and external monitor


About a second after you connect the external monitor, it should wake up and direct all output to the external monitor.  (It will only do this if the lid is closed, external power is connected, and something is connected to the video output port.)

Note that depending on the speed of your computer and other factors, it may take a little while for anything to show up on the screen.  Give it at least a minute if you don't see anything at first, before assuming something didn't work.