Macbook air booting from external hardrive stuck on startup wifi screen
You will not be able to boot from an original retail copy of Snow Leopard (10.6) as your Air requires 10.6.4 or later. Your old drive should boot your Air as long as the installed OS is 10.6.4 or newer.

If your old Mac is still accessible and the external drive has 10.6.8 or newer you could do worse than download the Mavericks installer and create a USB installer for your Air.

To do this first download the Mavericks installer from the App Store. This will be placed in your Applications folder. Once downloaded do not run it or move it.

Next open Disk Utility and erase an 8GB USB memory stick, naming it Untitled. If you name it anything else be sure to modify the command below.

Open Terminal and enter the following command-

sudo /Applications/"Install OS X Mavericks.app"/Contents/Resources/createinstallmedia --volume /Volumes/Untitled --applicationpath /Applications/"Install OS X Mavericks.app" --nointeraction


When complete you will have a full standalone installer, ready to roll.