What does belly up to the bar mean?
To stand at a drinking bar...a place that sells alcoholic
beverages. Usually, the bar, or surface that the drinks are served
on, is the approximate height of an ordinary person's stomach, or
"belly," hence to stand there is to have one's belly at the bar. In
another sense, it could mean to step up, to come forward. Or,
simply, 'Have a drink."