"I hope you too enjoy the game," he told fans.
Cavill, who shot to international stardom after playing Superman in Man of Steel in 2013, also took to social media to gush about New Zealand's scenery, saying Milford Sound was "magic".
Last month he shared a snap of the majestic Lake Wakatipu on a cloudy winter's day.
His M:I6 co-star and countryman Simon Pegg posted his own tribute to Queenstown on Instagram along with a photo of him with the iconic lake in the background.
The film's director Christopher McQuarrie shared some photos of spectacular New Zealand landscape with his followers via Instagram.
However, Pegg and Cavill's American co-star - Mission Impossible's leading man Tom Cruise - has been elusive during his time down under, staying quiet on social media while in the country.
But the superstar looked smiley while braving a cold snap in Queenstown for a photo last month, which McQuarrie uploaded to Instagram.
Earlier in June, a photographer spotted Cruise at Queenstown Airport.
He was reportedly there for a brief time while the helicopter he was onboard refuelled.
