Morpheus doesn't usually tire, Smith just kind of beat him. But as we find out, Smith is a whole different animal than anything else in The Matrix, other than Neo. Look at Morpheus' fight with an agent on top of a semi during the second film. That entire sequence lasts like fifteen minutes of him constantly being in action and he isn't really worn out.
Also wasn't Tai Lung's main form of attack using pressure points that didn't work because of Po's size? Morpheus seems to go for more direct force and redirecting a lot of attacks. Anytime Po tries to throw a punch, Morpheus can parry the blow and send him flying.
All of the fights you are talking about happened in the matrix. Even if he knows kung fu outside the matrix, he still has to breath air. Also in real life he is a 50 year old man, in good shape, but still. Pro wins this.