No, the feature you refers to is called
Target Display Mode, it's only available on some Intel iMacs with Thunderbolt or Mini Display Port. And they are very picky about their source, only Macs with the Mini Display Port can output to iMacs with Mini Display Port, same for Thunderbolt.
I believe Belkin has an accessory that will trick iMac into taking other sources. But the point is, you MacBook Air doesn't support it.
I will recommend getting a monitor, but if you really really want to use that screen, you can. Here's your option: Video Capture Cards. You can find them on Amazon.
Basically they are cards connected hopefully by USB or Thunderbolt to your Mac, and then transmit a video from a source connected to it. Higher-end ones will have HDMI 1080p input or H.264 encoding. Their main usage is to offload encoding job from CPU so their computer can focus on streaming content to the Internet. But you can however view that content locally.