What does "showing as much interest as a bee shows in vinegar cruet" mean?
Solution 1:
In popular imagining (and perhaps in real life), bees, wasps, and flies are attracted to honey but repelled by vinegar. Hence the saying "You can catch more bees [or wasps or flies] with honey than with vinegar."
In keeping with this understanding of the preferences of bees, the character Old Bryson in O. Henry's short story "One Thousand Dollars" (written circa 1905) views the affairs of Bobby Gillian with as little interest as a bee would have in viewing a cruet of vinegar. This is why, two paragraphs before the vinegar cruet simile appears, Old Bryson urges Gillian to take his as yet untold story off to the billiard room, where Gillian might find an interested audience.
The only complicating factor here is that O. Henry frames Old Bryson's reaction in neutral/positive terms (he is described as "showing as much interest as a bee shows in a vinegar cruet") rather than in negative terms ("showing no more interest than a bee would show in a vinegar cruet"). Presumably the author liked the misdirection and formal ambiguity of his formulation, which leaves to the reader the task of judging how much interest a bee is likely to have in a vinegar cruet, without explicitly stating that it is very little interest indeed.
Solution 2:
In the American Bee Journal June, 1916 issue, there is this item, contemporary to the short story you reference:
Texas, although second only to California as a producer of honey, made no exhibit at the [Panama Pacific International] Exposition, and had no literature on the subject. However, a delightful old Texas Colonel unofficially came to the rescue of the Lone Star State and assured us that honey-growing is one of its commercial industries; that from Beeville alone $1,000,000 worth was shipped in 1914; that the apiaries are all large, containing from 250 to 1200 colonies each; that the State annually appropriates $10,000 to fight bee-diseases; that the crop is from 75 to 100 pounds to the colony and consists of both extracted and comb honey, while unripe honeys find their way to the vinegar cruet. He attributed the success of the bee-industry to the dry climate and the abundant honey-flora, on which he was equally well-informed. [emphasis added]
It is likely that O. Henry’s contemporaries would have understood that unripe (thin) honey is (was) used as a feedstock for the production of domestic vinegar. From that we can guess at other possible bee reactions to the vinegar cruet and draw the appropriate parallels:
The bee understands honey production. A vinegar cruet is so far removed from the bee’s practical understanding that it is incomprehensible to the bee. Gillian’s concerns are uninteresting because the unique concerns of the rich are incomprehensible to those who are not.
Or, honey is the bee’s food store, which is taken to produce vinegar, so to the bee the vinegar cruet represents a theft. Gillian’s concerns are dubious because the rich gain by theft from those who are not.