According to this paper the Amami rabbit has apparently been successfully bred in captivity and one adult pair of these animals managed to produce 11 offspring (no idea on survivorship to maturity though) over a period of 5 years (1984-1989) at Kagoshima Hirakawa Zoo, in Japan.
Also, some information on the hubandry of the species while it was held at Kagoshima Hirakawa Zoo (quotes from the paper below):
"In the Kagoshima Hirakawa Zoo, the primary enclosure is 159 m", with a wall of 2.2 m height composed of 1.2 m of concrete and the other 1 m of wire netting. A secondary enclosure is 120 m", The primary enclosure has several oaks and pine. Captivity facilities have feeding and drinking posts. In the Central Institute for Experimental Animals, rabbits were reared in a laboratory (Matsuzaki et al. 1989). The animal room was kept at a temperature of 22 :t 2°C and a humidity of 55% :t 5%. Air was changed 10-15 times per hour with fresh air. Lights were on 0600- 2000 h. Ten rabbits (5 males and 5 females) were in reproduction cages (CLEA Japan, Inc., Model CR-II), 75 cm wide, 50 ern deep, and 35 cm high. Each cage was divided into 2 parts by a partition with an opening to pass through. One part of the cage was covered by an aluminum plate to keep it dark and was laid with dry grass for a nest; the other part was for food, water, and toileting. Insect nets were used to capture Amami rabbits in the enclosures of the Kagoshima Hirakawa Zoo when the animals were out of their burrows. To restrain an animal, the flabby skin from neck to back of the rabbit was held by the right hand, and the hind limbs held by the left hand. Juveniles were moved to a secondary enclosure to prevent parents from attacking them at the age of independence of the juvenile."
"In the Kagoshima Hirakawa Zoo, the diet was sweet potatoes, apples, and commercial pellets for rabbits; leaves of the fig Ficus erecta, akebi Akebia quinata, and other species, were seasonally supplemented (Sako et al. 1991). In feeding trials, 70 kinds of wild plants were eaten by P.funessi. Some Amami rabbits died due to infection by a tapeworm, Cysticercus pisiformis, carried by wild plants. At the Central Institute for Experimental Animals, grasses and weed were supplied just after introduction from the field, after which commercial pellets for rabbits (CR-3, CLEA Japan Inc.) and for pikas (CIEA-117) were the main diet, supplemented with sweet potato and apple (Matsuzaki et al. 1989)."
*Source : "Present status and Conservation of the Amami Rabbit (Pentalagus furnessi)", Yamada et al, 2000.