Yep Qatar. I like using booking.com so will wing it. I take it airport not far from ‘cbd’ as such?The Museum of Islamic Art is worth a look, as is the Souq Waqif.
What I did to kill a half day was to hire a (aircon!!!) car and driver and they just took me around, stopping where I wanted and waited for me.
Are you flying Qatar? Great airline, but beware their 'overnight hotel' deals, or rather the Oryx Rotana. It used to be handy to the old terminal, but now is not convenient and service is a shocker.