Born on 23 June 1844 at Otarihau, Hokianga, New Zealand, Gundry was the second child of William Richardson Gundry and his Maori wife, Margaret Rautangi of Ngaitupoto. He attended the Church of England Grammar School, Parnell and studied under the Rev John Kinder. A sketch by Kinder of Mercury Island, New Zealand, has the artist and Arthur in the foreground. Sketches by Gundry employed in the Magazine of St John's College, were highly regarded, as the work of 'Arthur Gundry, about 16 years old, a half-caste'.

Arthur left Auckland in 1863 for Sydney and then to Gravesend, Kent where he arrived on 1 June. In Britain he took classes at Mr Cary's Academy, Bloomsbury, and in the following year was admitted as a probationer and then a student at the RA schools. In the two years prior to his early death, he exhibited four works at the RA, and two works at SS immediately before he died.

Possibly in Cornwall to search of family connections, Gundry died of a fever in Penzance at the age of 24. This was well before the influx of artists that became known as the 'Newlyn school', however it is clear from the title of a painting exhibited at the RA in the year of his death, that he was already engaged in genre paintings (or story-telling pictures). The work referred is Making her way from Madron Church Tower to Penzance Market.  The whereabouts of this painting is unknown, and to date no images of his work have been found.

His lodging address in the Penzance area was 1 Aubrey Villas, and Gundry is buried in Paul Churchyard. (Paul Burial Registry)

His death, reported in the Daily Southern Cross of 29 July that year, said "His artistic talents were of a very high order, and had he lived he would have undoubtedly attained great eminence in his profession. He was greatly esteemed by a large circle of acquaintances both in this country and England."