Bougainville is reached by air via Buka Airport, which has flights to and from Port Moresby operated by Air Niugini. The flight takes about two hours. There are no direct international flights; you will transit through Port Moresby regardless of where you are coming from.
There is no regular passenger boat service from the Solomon Islands, though occasional cargo vessels make the crossing. The Solomon Islands chain is geographically close — the border runs through the ocean just southeast of Bougainville — but the political and logistical gap between PNG and the Solomons makes overland-style travel between them impractical.
Once in Buka, road transport across the narrow channel to main island Bougainville is via short ferry. Roads south from Buka to Arawa and beyond are rough and may be impassable in the wet season (roughly December to April). Local boats along the coast are often the more reliable option.