I agree with Polonius -- those are the top two sites in Doha. Others of moderate interest are the Pearl, Qatar's man-made island; the Villagio, a shopping mall; and, in the evening, the Sky Bar at La Cigale hotel, which is also within walking distance of probably the best area (Al Saad and Al Nassr Streets) for local food.

Taxis are very cheap. A taxi to souq waqif, which is close to the airport, shouldn't be more than QAR 20-25 (about USD 5.50-7.00). You can also catch a taxi at the souq, or at any shopping mall or many hotels, back to the airport. But that's about it -- beware that taxis are in short supply in Doha, so you have to plan in advance to get back to the airport.

If I had 16 hours, I would certainly get a hotel room. One downside for the OP is that the stay spans two nights, unless you could perhaps kill some time and convince a hotel to check you in early. Or, at worst, leave your luggage at the front desk and spend time at the pool, beach, or spa before you check in, for which the Sharq Village, a Ritz-Carlton property next to the airport, is a good choice -- although not cheap. Other options close to the airport are the Marriott (on the beach) or old MovenPick.

zhoux, no way would I spend 13 hours in the premium terminal, especially the business class side. There is nothing to do, no sleeping area, one channel of television, and scant reading material. There is a shower and relatively substantial, albeit mediocre, food.

Of course Doha is not the most interesting city in the world, or even the Gulf, but if you wouldn't otherwise go there, why not take advantage of a day and see the place?