Sultan Thaha Airport, as I mentioned before, is a small airport with only 8 flights in a day (no flight at night). And for sure, Garuda Airways doesn't have flight to Jambi City. You can flight with garuda Airways to Palembang or to Padang then use land transportation to Jambi City.
Once I asked the Novotel guy, I asked him what kind of tourism attractions that Jambi City has. Actually there is no tourism attraction inside Jambi City, all tourism spots are outside the city: Jambi Museum (km 12 Kotabaru), Candi Muaro Jambi (Kabupaten Muaro Jambi), Danau Gunung Tujuh (lake of Seven Mountain, Kerinci Seblat National Park area), Kerinci Volcano and Kerinci Seblat National Park.
But for shrimp lovers, find the best shrimp in town at Sari Wangi Restaurant, near Sultan Thaha Airport, Jambi. You'll love it.
Related link: http://www.muarojambi.go.id