Using an external monitor will work via the Thunderbolt port. (using a Thunderbolt to HDMI, or even Thunderbolt to VGA adapter) Of course then you need to connect the two via Wi-Fi, or ethernet and "share" the hard drive.
Using it with an external monitor also makes your MacBook Air a second "desktop" computer, so it doesn't need to be thrown away. Many Macs find themselves being "handed down" to family, or friends who have no computer, or an even older one. They would be happy to have a good working MacBook Air, even if it is only usable as a desktop via an external monitor.
I have an old PowerBook (forerunner to the Mac Book series) in my network that has a bad battery charging system, so the battery won't hold any charge. I leave it connected via its external power supply/charger and it works fine as a portable work station doing things that an old G4 CPU running OS 10.4, or OS 10.5 can handle. It comes in handy for using old "Classic" apps and games that aren't supported under the newer OS versions and converting old file formats to a newer format that can then be used on the newer Macs.The first technical book I wrote under my own name was done on a Mac Plus using the old ClarisWorks text applications, which used a format that the newer text apps won't recognize. I converted it to a newer format and was able to put it in my new work station, so that I could update it and release a "revision" for sale.
The best and easiest way you can connect the old one to a new one is with a thunderbolt cable:
1. Connect the two Macs with a Thunderbolt cable. This won't work unless they are both Macs and your old one is 2011 or newer, designated as MacBookAir4,1 or higher. (Under the Apple/About This Mac/System Report/Hardware Overview and look for the "Model Identifier") Older models do not have a Thunderbolt port, but instead have a "Mini Display port".
2. Start the new one and wait until it is fully started. Then hold down the "T" key on the keyboard of the old one as you start the old one. Since you can't see the screen, simply hold the "T" key down until it appears on the desktop of the newer Mac.
NOTE: By default, Macs now come with no hard drive icons displayed on the desktop. To make them appear when the computer is running, click on "Finder" (top of the screen), select "Preferences", Select "General" and check the boxes for anything you'd like displayed on the desktop. You can deselect them later again if you wish.
3. It will display as an external hard drive on your new Mac's screen after it starts completely.
4. From there you can drag and drop any files, or even use "Migration Assistant" (In Applications/Utilities in the new Mac.) if you want to completely move everything including the old OS and all of its settings.
BTW: The same "Target disk" mode I just described will also work with older Macs that have a Firewire port by substituting the Thunderbolt cable with a Firewire cable. Makes file transfers so much simpler and quicker than "sharing".