Sorry, but I'm not aware of any user-available utility that will allow you to test an AirPort base station regardless of the model. Typical failures of most networking electronics are power supply-related.
However, before writing off the Express for potential hardware failure, you may find that the issue is more related to signal quality. The primary effect to this is Wi-Fi interference.
To get an idea of the signal quality you will want to make Signal to Noise (SNR) measurements at various locations where you want the wireless client to connect to either or both of your Express base stations. The following is a simple test you can do right from your MacBook Pro. Note: To get accurate measurements you will want only a single Express operating at a time.
For the test I mentioned, you will use the built-in OS X Wireless Diagnostics, as follows:
- Option-click on the AirPort icon on the OS X menu bar.
- Select "Open Wireless Diagnostics..."
- Enter your Mac's Administrator credentials in the next window.
- On the Wireless Diagnostics window DO NOT click on "Continue." Instead, go to the diagnostic's menu bar, and then, select Window > Utilities
- Start the test with the laptop near the wireless router. Note the values for RSSI and Noise. They should both be negative and measured in dBm.
- Now take the laptop to the other locations that you want to access the Internet in your home. Different rooms, different floors, etc. Again, note these same two values at each of these locations.
- When you have finished collecting the data, you will want to enter them into the following formula for Signal-to-Noise Ratio (SNR). SNR = RSSI - Noise. For example, if your get the following readings: RSSI = -61 dBm and Noise = -84 dBm, SNR = (-61) - (-84) = 23 db
To get decent wireless bandwidth performance at a location you will want the SNR value to be 35+ dB.