Having just operated to Baku in the last couple of weeks a couple of points.
Prior to Baku travelling east MSA reaches up to 18400' so depending on where it happened choices of diversion are limited after a descent due to decompression.
Take a look at Tblisi Odesa Simferopol or Ganja. Where wold you rather go. Again depending on where it occurred there is a lot of terrain and distance between Baku and other airports outside this area.
As for making a pa? In an emergency descent on my little airbus there is a pre recorded announcement not sure what happens on a 380. I sound like a Dalek through the oxymask so it won't be reassuring to hear from me early on.
So it's an emergency descent due to probably slow decompression with hi terrain all round. We are trying to prevent the cabin pressure rising above 14000'. If all that comes out of the blue regardless of how good your situational awareness is at that point in the cruise it is I can assure you very very busy up front. The passengers can wait.
It's a good diversion airfield with relatively decent facilities. The new terminal according to ground staff opens in may but the old one is ok albeit without a huge amount of eateries and shops. There are a couple of hotels on the airport as well.
All safe , well done the crew .