Done it a few times now - occasionally have a "taxi" experience as I'm the only passenger for the entire route, and when it's slow due to traffic, a taxi will get caught up in the traffic too. Only downside is that the bus stop is on the opposite side of the airport to J/F check-in!
Buy your ticket in baggage reclaim near to the exit, it's 10 riyals for 2 journeys by smartcard (or 20 riyals for 24 hours). You can also pay 10 riyals cash to the driver for a single journey. The drivers are supposed to stock the smartcards but even if they do, they may not sell them to you as they prefer to pocket the cash (the bus company wants to know if this happens, but will you really bother to report them?) One time I paid in coins - wonder how many QR-loving flyertalkers have ever touched a Qatari coin
Also - nice photos
hkskyline - thanks for taking them so I didn't have to, I basically did your trip in reverse recently.
I also took Uber one early morning from the city and it only costed QAR 21.