There is no priority lane upon arrival. There is (although I here it's hit & miss) a priority service upon departure. Its handled by a 3rd party (Coral Executive) & pax are accompanied to a dedicated counter/s at the far left side of the emigration counters. The service is also available as a paid service for Y pax.
And no, for those that don’t know, it is definitely not always a quiet/calm experience arriving or departing from HKT (depending on time of day it can be extremely busy) so a service like this is indeed well worth it.
This is based upon my experience with EY & QR but I am guessing it will be similar for EK too.
I've just come back from HKT flying business with Dragonair (it didn't offer any fast track services) we used the Coral Executive immigration services. I can highly recommend both arrival and departure as the new airport is still a shambles with large queues. The Coral lounge is ok but miles better than the old terminal. Its around 1500 thb each way.