Keflavík International Airport: Rent a car at the airport: it is the genuinely useful transfer here, roughly 2¼–2¾ hours to Hvolsvöllur via Reykjavík and the Ring Road, depending on weather and stops. There is no airport-to-Hvolsvöllur shuttle worth treating as a default. Without a car, take an airport coach to Reykjavík, then a scheduled south-coast bus; expect a connection-dependent, most-of-a-day journey rather than a slick transfer.
Reykjavík BSÍ bus terminal / Mjódd bus hub: For car-free travel, use a scheduled south-coast service toward Hvolsvöllur/Vík when it operates; schedules are seasonal and limited, so verify the exact stop before committing to accommodation. Strætó route 51 reaches Selfoss from Mjódd, but it does not solve the final leg to Hvolsvöllur on its own. By car, Hvolsvöllur is about 106 km east of Reykjavík, typically 1½–2 hours on Route 1 in decent conditions.