Oslo Airport Gardermoen  OSL is the closest airport with bus transfer several times each day to and from Hønefoss. Look for the buses marked "Askeladden"
Trains pass through on their way to Bergen, Oslo and Drammen. About two hours travel time from Oslo. The train station is located just outside of town, within walking distance from the town centre. Just follow the foot-path downhill.
Buses leave every hour from Oslo Bussterminal, until late in the evening. Night-buses on the weekends. The trip takes around one and a half hours, and is worth it for the view alone. You will get a magnificent view over the great Tyrifjord Lake, one of Norway's biggest inland lakes. Make sure you get a seat on the left side of the bus when travelling from Oslo, to get the best view.
Askeladden reiser  has a regular transport from Hønefoss to Oslo airport Gardermoen  eight times every day from 05:30 in the morning to the last return from the airport at 10:45 in the evening, except for saturdays. Check the schedule under the link "Gardermobussen". 
As in all of Norway, there are few real budget accommodations available. The cheapest option is to simply camp out in one the many woodlands that surround the town. One such place, within short distance from the town centre, is in the Kartverk-forest. Take a bus from the centre to the stop "Kartverket" and simply walk into the forest from here. Always remember, if you're travelling Norway on a limited budget, that you can camp anywhere you want as long it's not in someones back-yard or on a cultivated field.