Being Iceland's northern capital doesn't mean Akureyri is a huge metropolis. Although it can get a bit hilly leaving the waterline, walking distances are reasonable for a fit traveller.
Akureyri has a public bus system, called SVA, which is free for all. The buses are not very frequent--each line has a bus every hour or so. The buses stop running at 23:00 on weekdays and only run between 12:00 and 18:00 on weekends and holidays, but when they're free why complain?
BSÓ (http://www.bso.is/) is the only taxi company in town, and they have a taxi rank in the centre of town.