I'm going with this under ComicBookResources Rumbles rules because they are the best rules to go by.
Working on the assumption that Cap has his shield and Bats has his utility belt, I'd go with Batman by a small margin. The belt contains grenades, sonics, projectile weapons, sedatives and a whole host of other things that Batman has proven he's more than capable of deploying mid-fight. Cap is deadly with the shield and would be able to take a very solid minority of the fights, 4/10 to give number, but the belt is too versetile in my opinion. Bats techs his way to victory.
Conversely: In straight up hand to hand, Cap takes Batman. Stronger, possibly slightly faster, unlimited stamina and just as much experience and skill. Even if he can't find a way to put Batman down, which I have every faith that he can, eventually Batman will tire and Cap won't. War of attrition baby.