Richard Arnst was the eighth of thirteen children born to Hermann and Catharina Arnst. The family lived at Tai Tapu near Christchurch. He, and his brothers Jack, Herman and Bill (William), became champion cyclists both on the road and on the track. Richard and Jack were placed fourth and third respectively in the 1903 Timaru to Christchurch road race which was over a distance of . Jack subsequently won, in record time, the 1903 road race between Warrnambool and Melbourne over a distance of . Some of the credit of this win was due to Arnst's unselfish pacing of his brother. Richard gained 5th place in this race in the second fastest time, that was inside the previous best time. The brothers returned to Australia in 1904 with Richard finishing in 10th and Jack well back in 25th. In 1905 Jack could only manage 76th with Richard in 77th.