Would a 19,000 ft descent in 6 minutes feel noticably different to passengers?
From the physical side, you can not measure / feel if you are at a constant altitude or if you are ascending / descending with constant speed. What you can feel is the change of speed in any direction, and acceleration acting in an unexpected direction. This causes the usual sensations:


Accelerating / braking on the runway: You are pressed into the seat / belt.
Weird feeling right when the aircraft becomes airborne: The aircraft accelerates upwards, which presses you into the seat with more than 1g. This higher g-force feels strange.
While climbing, the aircraft still accelerates, but the more strange feeling comes from the fact that the force doesn't point down to the floor of the aircraft, but more backwards. Your eyes see that you are sitting upright in the aircraft, but it feels more like lying.
At some point, usually when the flaps are retracted, the climb rate decreases suddenly and it feels as if the aircraft falls. This is because the aircraft accelerated downwards - tough it still climbs, but at lower rate.
Usually, you don't feel much during a coordinated turn. Maybe you feel a little heavier, but when you look out of the window and don't see horizon, this again gives a strange feeling.


Of course, the passengers of that flight did have several sensations. right after the incident, the aircraft for sure has done some unusual movements, plus what the pilots did. 

The reference #7 of the Wikipedia article contains this diagram on page 36:



Directly after the incident, the aircraft climbed a little and the speed brakes were deployed (50%) to reduce speed. This was for sure noticeable by the passengers. And I guess the aircraft started to descend faster than usual, so the sensation of "becoming lighter" will have been much stronger/longer than usual.
After this, the aircraft descended with a quite constant rate for several minutes. This doesn't cause the feeling of being lighter. Maybe the attitude was unusual which means gravity doesn't push you vertically into the seat.

Later, when the aircraft reached 10,000ft, it accelerated upwards to stop the descent. At this moment, the passengers felt heavier.

(If you have a closer look, the descending part isn't a perfect straight line. This wasn't a smooth ride, but as said, a constant descent rate doesn't cause a sensation)

Of course, this is just about the sensation of acceleration so far. The passengers for sure have noticed the drop in cabin pressure, drop in temperature, more noise and so on.