Styris made an unbeaten 58 after Astle had scored 51 as the Kiwis limped to tea on 216 for seven after collapsing to 69-4 before lunch.
The pair added 71 runs in less than 18 overs before Dwayne Smith struck three times in the last hour to put the West Indies back in control.
The tourists had no hesitation sending New Zealand in first on a temporary drop-in pitch at Eden Park that suited their bowlers and the decision immediately paid off.
Paceman Ian Bradshaw captured the vital wickets of Stephen Fleming (14) and test debutant Peter Fulton (17) as well as running out rookie opener Jamie How (11) while Jerome Taylor dismissed Hamish Marshall (11).
How departed in the 10th over in unfortunate circumstances, run out while backing up at the non-striker's end when Bradshaw flicked a Marshall drive on to the stumps.
Marshall, promoted to opener after Lou Vincent was surprisingly dropped despite being in great form, also went for 11 when he chipped a full-toss from Taylor straight to Fidel Edwards at mid-on.
Fulton was caught by Daren Ganga at gully off Bradshaw after taking nearly an hour to make 17 while Fleming departed in the last over of the morning session, caught by wicketkeeper Denesh Ramdin.
Astle and Styris launched the New Zealand comeback after lunch. Astle hit 10 boundaries in reaching his half-century while Styris struck 11 fours in his 58 not out.
Smith had Astle caught behind with the total on 140 then bowled Brendon McCullum for a quick 19 off only 15 balls and had Daniel Vettori caught by Chris Gayle for six.