The New Zealand opener made a brilliantly-paced 83 on a slow pitch, just when his team-mates had been struggling for runs against Ireland's disciplined seam-spin comination.
Fulton, a middle-order batsman opening the innings in the World Cup, had made only 62 in his last three matches before regaining form with his sixth half-century which contained nine fours.
He did well to keep his end intact for a long time when wickets were falling at regular intervals from the other. He put on 54 for the fifth wicket with Jacob Oram, the second-highest stand of the innings.
It was only a late flourish by wicket-keeper Brendon McCullum (47) and James Franklin (34 not out) that helped New Zealand reach a respectable total, with the last three overs yielding 45 runs.
The pair put on 71 for the eighth wicket, with McCullum hitting a six in the last over that smashed a window-pane in one of the boxes. Franklin struck one six and three fours in his 22-ball knock.
Ireland had earlier done extremely well as they never allowed New Zealand to build a partnership, with seamer Dave Langford-Smith and off-spinners Kyle McCallan and Andrew White grabbing two wickets apiece.
Skipper Stephen Fleming (10), Hamish Marshall (16), Scott Styris (10), Craig McMillan (22) and all-rounder Oram (20) all got starts, but none could play a long knock.
New-ball bowler Boyd Rankin provived the breakthrough in the sixth over when he dismissed Fleming, who was caught cutting at point by William Porterfield.
Langford-Smith then struck twice in the space six overs to put New Zealand on the back foot, removing Marshall and Styris. He finished with 2-41 off 10 overs.
New Zealand found runs hard to come by when Irish skipper Trent Johnston introduced his spinners into the attack, as McCallan and White kept a tidy line and length on a slow track.
McCallan was rewarded for his discipline when he got rid of McMillan and well-set Fulton, who was adjudged leg-before. The off-spinner conceded just 35 in his 10 economical overs.
New Zealand 263 for 8 in 50 overs (Fulton 83, McCullum 47, Franklin 34 n o, McMillan 22; McCallan 2 for 35, Langford-Smith 2 for 41, White 2 for 45) Vs Ireland (yet to bat).