The remnants of Cyclone Sandra delayed their arrival at the Basin Reserve until lunchtime. But in the predictably short window of opportunity available to England, before rain wiped out the afternoon and evening sessions, the tourists could make no significant headway.
After James Anderson had Fulton fencing a catch to slip in the fifth over of the morning, Kane Williamson (51no) helped New Zealand move from 77 for one to 153 for two.
That put the Kiwis just 58 runs adrift of England, following on - and with Monday's weather forecast much worse than Sunday's, there is a high probability that this three-match series will head for its final instalment in Auckland still stuck at 0-0.
England had 29 overs in which to make inroads, on the back of Stuart Broad's six for 51 on Saturday and centuries from Nick Compton and Jonathan Trott before then in a total of 465 all out - which also owed much to Matt Prior's expertise, batting with the tail. But save for Anderson's sole success, they got nowhere.
Opener Fulton had used up 114 balls and was just five short of his second half-century of the series when he succumbed, ending a stand of 56 with Williamson.
On a pitch showing precious few signs of deterioration yet, Williamson was in even less of a hurry and duly dug in again for an unbroken half-century partnership with Ross Taylor.
Monty Panesar got two deliveries to turn sharply in his first over, and found the edge of Taylor's bat too but saw the ball fall just short and wide of slip. Taylor also twice played and missed early on as he looked to attack Finn.
But otherwise, there was minimal encouragement for England, and number three Williamson completed his 50 when he pulled Joe Root for his seventh four from the 155th ball he had faced.
He had done enough already to dent the tourists' ambition, and four subsequent hours of intermittent rain could only have further dampened their spirits.