Is it possible and relatively easy to boot a 2013 MacBook Air from an external drive?
It should work, depending on the model of MacBook Air an external USB drive will be able to boot the system but it will likely be slower than an internal drive on the PCI/SATA bus. You should also have a Thunderbolt or mini display port for connecting a monitor. 

So it should work. Whether or not it will perform acceptably is difficult to say.