Last summer I got an AirPort Express from one I know in the states. It turned out that it was completely dead when you plug it in the wall. The first thing that hit my head was that the Power Supply was faulty. A quick Google search confirmed that many others have had problems with the AirPort Express PSU.
Some pictures of how I got it back to life:
Since I used a linear regulator at first, I had a huge loss of power (5V to 3.3V ~1A) over the regulator. And yes: It did get very hot. After some initial testing I replaced it with a small buck-regulator (in the lid under) that does the job: