Daimaru
Aside from the many restaurants clustered herein, there's a Starbucks 8AM-9PM Su-Th, to 10PM F-Sa for coffee hounds and a beer garden on the 8th floor rooftop garden late May to early September, 5:30PM-9PM with bar food and all-you-can-drink for ¥3300.
Uomasa Honjin
While you're drinking, order a la carte from their menu of sashimi including fugu and whale, nabe, and various fried items. And if you'd like to mix your poisons, ask for the fugu fin sake.
Buzenda-cho è±åç°çº is the local red-light district, packed with karaoke boxes, hostess bars, and restaurants, a short walk east of JR Shimonoseki Station. Bars are also clustered on Hisashi-guchi, near Green Mall.
Kanmon Wharf
Beer garden by the sea, from late April to the end of August. Admission includes plenty of snacks.