Who would win this battle? One would think Captain America would win because he actually has a boost in enhanced strength, speed, and stamina due to super soldier serum and all Bats has is his normal strength, fighting prowess, and keen intellect. However, add to that Batman's sneaky ninja tactics, vast arsenal of weapons and armored suits, then the advantage falls to Batman. After all, although he is only human, Batman has taken down many superhumans using only gadgets, strategy, and intellect.
Cap may be a super human, but he is susceptible to many things normal humans are. A potent gas or poison projectile designed specifically for Cap's unique body would end the battle fairly early.