It was Rutherford's second highest score after making 171 on debut at the same ground against England in March.
His half century came up in 60 balls and included nine boundaries. Fulton was a little more circumspect, preferring to punish the bad balls when he got the opportunity, with his first 20 runs all coming in boundaries.
The tall right hander was on 31 at the break, having been joined by Aaron Redmond (four) after Rutherford was well caught by Narsingh Deonarine in the outfield trying to hit off-spinner Shane Shillingford out of the ground.
West Indies have had a disrupted buildup with most of the squad arriving only at the weekend after their tour of India. Several players are also battling illness and fatigue after their short turnaround.
Their destructive opening batsman Chris Gayle was earlier ruled out of the three-match series with a hamstring injury while his replacement Kraigg Brathwaite was also still waiting to join the team after visa issues.
Sammy wasted little time in opting to bowl but his two front-line fast bowlers Tino Best and Shannon Gabriel failed to exploit the conditions by bowling too short of a length.
The only time Rutherford and Fulton were troubled was when they pitched the ball up, though those deliveries were few and far between with Sammy - the third pace option - the most consistent of the trio.
He produced a half chance in the first session when Fulton nicked a delivery that would have gone to third slip had the batsmen not forced the West Indian captain to adopt more defensive field placings.
Shillingford, who had been under a selection cloud following bio-mechanical analysis of his action in Perth, was introduced about 30 minutes before the break and tricked Rutherford into attempting to hit his second six shortly before lunch.
The second test is in Wellington from December 11-15, with the final game in Hamilton from December 19-23.
(Reporting by Greg Stutchbury in Wellington; Editing by Peter Rutherford)
