The 11,800-capacity stadium is one of two new venues being constructed in Milan before the 2026 Games.
Santagiulia is due to host many of the biggest ice hockey fixtures at the Games, including the men's and women's gold medal matches.
But days before the Games, most roads remain closed around the arena while inside construction materials and rubbish litter the floor.
Many areas of the arena, including hospitality boxes as well as food and drink stalls, are unfinished, while there are no seats in the press area.
While core elements like electricity and running water are functioning, lifts and toilets remain covered in plastic wraps and are scruffy in appearance.
"No-one's experience will be tainted by anything that needs to be painted after the games," said Dubi. "Anything that is public facing will be top.
"Do we still have works to do? Yes, cleaning absolutely. There are still works ongoing, frantic as you say but it is to make it a great venue."
Dubi praised the work done on the venue, which has been beset by delays and controversies - including a boycott warning from the National Hockey League (NHL).
The rink in Milan, approved by the International Ice Hockey Federation, is shorter than the minimum requirement in the NHL, leading to suggestions there could be an increase in high-speed collisions. The quality of the ice has also been questioned.
The NHL said it was "pleased" with the test event at the arena in January, although that did not pass without incident. During the event there was a short delay while a small hole in the ice had to be repaired.
Santagiulia is one of two ice hockey venues for the Games, with the Milano Rho Arena staging some group-stage matches.
The Rho Arena, a temporary venue constructed in the Fiera Milano exhibition centre, has a capacity of 5,800 - 6,000 fewer than Santagiulia.