1) To and from the airport
The international airport (Keflavík) is served by two coach companies that ferry visitors to and from Reykjavík: Airport Direct and Reykjavík Excursions. They both transfer travellers from the airport to the central bus station in Reykjavík, and from there will drop you off at your hotel. You will inform the company of your hotel when you make the online booking. The airport is around 50 km away from Reykjavík city centre.
Taxis from the airport into the city are always more expensive (~25,000 ISK, see e.g. here) than the coach even when shared, as far as we are aware.
It is also possible to take public transport (bus 55), but the time table is very limited: https://www.straeto.is/uploads/files/R-55-pdf-a75790e6da.pdf
