They operate daily 6 am-1 am. If you want to go to the airport, you should call the day before your departure. You can get there by tram number 4, the route of which includes Aleksanterinkatu. You can get to L nsisatama by bus number 15 from the bus station. You can get all necessary information, timetables and route maps free from HKL ticket offices.
Local buses dominate the public transport system.
They will probably get you to your destination, although the Metro and local trains are faster. There is just one Metro line but local trains take you to the suburbs. Regional tickets allow you to travel in Helsinki, Espoo and Vantaa. Tickets remain valid for one hour after the time stamped on the ticket on entering. Remember to stamp your ticket at the special machine when entering the vehicle.