"belly up" is a verb that means: to move close or next to.
For example: belly up to the buffet table and help yourselves. "Belly up" means g to the buffet and have some food, make yourselves comfortable here (like in a party or in a restaurant).
In this one sentence it literally means: come so close to the bar that your belly actually touches it. As if you were going up to a bar to get yourself a much-needed alcoholic drink (yahoo).
It could also mean "cair de cabeça" referring to the bar itself.
"Simply belly up to the bar, and give your order to a bartender."