No, the Qatar National Library isn't currently by the main post office. Reason being it's under construction, and from what I understand they're experiencing delays. I believe they're having the same issues as the other ubiquitous construction projects here--running out of cement, permits, etc.

Mr Niceguy's directions are good-- check out a map and you'll see what he's talking about.

Here is an alternative: how about the Education City libraries? Georgetown, I know, offers public membership. They have a nice collection, guys. I just read John Grisham's newest courtesy of them.

There's 38,000 books, and they are buying at a furious pace. Their most-used vendor, for instance, they receive anywhere from 3-400 books a week. This is very large-scale acquisitions...

You pay a 50 riyal deposit and can check out 6 items at a time, for 2 weeks tops. No audio/visual stuff, or course reserves. No ILL services available. They buy magazines that people want to read. You may sit and read. http://library.georgetown.edu/qatar/

I don't know about the other schools... well, I do kind of. Cornell doesn't offer public membership. But TAMU-Q, Carnegie Mellon, and VCU-Q... I don't know about their public borrowing policies. You'd have to ask. TAMU-Q has some fiction stuff. VCU-Q and Carnegie Mellon don't really so much. Your best bet is Georgetown.