We do for the Romans. The Roman Military text "On Military Matters" by Flavius Vegetius has survived intact to the modern day and it includes some detail as to the training requirements of Rome.

In terms of a fitness test you had to march 20 miles in five hours in military step and 24 miles in five hours in full step. He states that any faster then that is running and not what you want. You would also have a swimming component as well as vaulting onto a wooden horse. You had to do the March and the vault in full gear which weighed around 60lbs. There was also a jumping component since it was thought that soldiers needed to be able to jump over ditches. The rest of their training went onto military arms and other tasks that were not purely physical.