I went out in my own by taking public buses. There are two buses which goes from Airport to bus terminal in Doha and it's just next to the Souq. Bus 747 and 109. The trip takes about 25 to 30nins with no jam. I thought the souq will be interesting during Ramadan but unfortunately, it's kind of too touristy. Had a better experience in Amman.

So if anyone has long transit, can use the buses. Quite a good experience. After arrival, turn right to the end and exit there are coach berths and look for 109 or 747.