What does “velvet-rope-poor” mean?

Given the text says Azerbaijan wants to be the world’s next party capital, it's probably a reference to the velvet rope you often see outside swanky party venues where people are queueing to get in...

Saying they're currently velvet-rope-poor doesn't mean they can't afford velvet ropes - it means you won't see scenes like the above very often in Azerbaijan, because they don't have that kind of society. But they want it, because they want to become more "Westernised".

The construction XXXX rich, [but] YYYY poor is often used to contrast the abundance of XXXX with a lack of YYYY. It needn't necessarily have anything to do with economics. For example...

Computers and the Internet have created a world which is data rich but information poor.

...is a sentiment which has often been expressed using the rich/poor juxtaposition.