The nearest airport is Sapporo's New Chitose Airport, served by Donan Bus direct to Noboribetsu (¥1,170). You can also take a train directly from the airport.
Noboribetsu is located on the JR Muroran Line between Oshamambe and Tomakomai.
Several limited express trains stop at Noboribetsu: The Hokuto and Super Hokuto, on runs between Hakodate and Sapporo, and the Suzuran, on runs between Sapporo and Higashi-Muroran.
From Sapporo, Noboribetsu can be reached by limited express in about one hour at a cost of ¥4160. The run from Hakodate to Noboribetsu is about 2 1/4 hours at a cost of ¥6500.
Noboribetsu is also served by overnight trains: The Cassiopeia and Hokutosei trains to and from Ueno Station in Tokyo, the Twilight Express to and from Kyoto and Osaka in Kansai, and the Hamanasu Sapporo-Aomori express (the Aomori-Sapporo Hamanasu does not stop in Noboribetsu).
If you're looking for a cheaper alternative than the train, Chuo bus and Dounan bus offer cheap, ¥1900 yen (one-way) trips to Noboribetsu in about 1 hour and 40 minutes. Boarding happens at Hakodate Station and Chuo bus terminal [and schedules can be found here (relevant station