The best hotel in my experience is Lacigale but it is likely pricey.

As far as what to do, depending on the day of arrival and how quick you get out of the airport, you could try to catch Souq Waqif. Their hours are http://www.souq-waqif-doha.com/opening-hours/

I don't think you will catch the Islamic Museum of Art (they close at 8 on weekends and 5:30 on weekdays) http://www.mia.org.qa/en/visiting/opening-times
or the National Museum (http://www.qm.org.qa/en/project/national-museum-qatar)