How can a passenger determine airspeed without a handheld GPS?
It is impossible for a passenger to determine the airspeed to any reasonable accuracy (otherwise Birgenair 301 and Aeroperu 603 would not have crashed).

From my personal experience, it is not possible to receive GPS signal using a handheld-device in the passenger cabin. Even if it works, winds aloft at cruising altitude are often in the range of 100~200 knots. That throws the calculation completely off even if ground speed is known.

The only reliable way is to refer to the entertainment system, which obtains its data from the cockpit and is therefore accurate. However, the interface is usually programmed to display measurements in other units; the indicated airspeed has to be calculated from the displayed values manually.

Here are some ways you can guess the airspeed to a certain extent:


If you do not notice any significant airframe vibration, you can
assume the airspeed is somewhere between the stall speed and the max
airspeed.
If you can tell the flaps position from observation, you can also correlate that with the usual flaps extension speed of the model
you're flying, which should give you ~20 knots of accuracy.
If you are familiar with the SID or STAR of the airport, and you know planes normally pass a certain point at a certain speed.
If you have a handheld ATC receiver, sometimes ATC will issue speed instructions to planes during approach.