Nong Khai is a great place for cycling due to the limited traffic and the nearby countryside. Many guesthouses let you use their bikes for free. If you need to hire one, Khun Noui, who sets up on the roadside across from the entrance to Mut Mee, has reliable bikes (50B per day) and motorcycles (200B); if he's not there, any of the túk-túk drivers parked nearby will give him a call and he'll be right over.
You can find metered taxis at the bus station and the bridge. Generally, people agree on a price rather than use the meter. A túk-túk between the Mut Mee area and the bus station or the bridge will cost about 60B and 100B respectively for two people.