Longyearbyen is a small town, and walking around downtown Longyearbyen is a wonderful experience. There are also two taxi companies ready to take you around town, into the valley, between the town centre and Nybyen, and to the airport. Longyearbyen Taxi and Svalbard Bus and Taxi are at your service 24/7.
There is an airport shuttle bus that stops at all the hotels and guesthouses in town. During the Polar Summer, you can borrow bicycles at the Tourist Information Centre, while Arctic Autorent is the town's car rental (located at the airport).
Source: Visit Svalbard