Business buzzed according to the urban dictionary, means to only have a drink or two or possibly three (depending on a person's tolerance and what they're drinking), because the person has to remain mentally capable and functional to do business, to meet and converse with others during a lunch/dinner meeting or at a party/gathering... or possibly to go back to work after lunch.
The urban dictionary only gave two sentences using the phrase and no explanation/definition, so this is my definition.
(Business buzz) is an acceptable level of intoxication for business situations. ~ Inebriation, but not falling-down drunkenness.
Therefore, "business buzzed" means feeling and showing the effects of alcohol consumption, but still able to function (albeit at a reduced level). "Buzzed" is a slang term. Random House Unabridged Dictionary defines it as "[having] a feeling of slight intoxication". Business buzzed suggests that slight intoxication is acceptable for the conduct of business. Some employers might disagree.
"Buzzed" here means intoxicated. "Business" modifies it and describes the degree of intoxication: "a level of intoxication at which it is still possible, or acceptable, to work, or do business".
It would be the equivalent of mild intoxication, generally, but as Yankee says, it depends on one's tolerance and ability to function under the influence of alcohol.
The sense of the quotation is approximately: "I wasn't able to get too drunk, as my boss was present".