First I was going to say in Honolulu, then I was going to expand to Oahu, but I figured might as well include the whole state.
I'm guessing that the House Without a Key probably has a good shot at this honor, although it has been rebuilt at least once, but references go back to at least 1925. Wo Fat in chinatown went back to 1882, but it closed a few years ago.
Oahu Country Club goes back to 1906, and the Pacific Club goes all the way back to 1851 in one incarnation or another, but those are private clubs and I'm not sure they could be included in the category, even though they serve food. And while the Moana was opened in 1901 it doesn't appear to have a continuously operating restaurant from anywhere near that time, nor does the Royal Hawaiian opened in 1927.
Any other nominees for oldest? Or even very old? Seems in this town anything more than 20 years old is old, although there are a few places like The Willows that go back as far as the postwar period.
House without a Key 1925
The Willows 1944