You may find this thread about Sparta of interest.
But more generally, there seems to be an implication about trying to build muscle mass for its own sake in your question. As the various sources compiled by Vegetius, a late Roman writer, suggest stamina was far more important than raw strength or size.
This is not sourced, so caveat emptor, but it is based on my study of Roman military history (such as it is.) Success in infantry combat in most cases is not so much a function of individual confrontations as it is a result of the ability of individuals to move and fight in a coordinated fashion. So if you have a certain amount of time to dedicate to training, you're much better off focusing on exercises to foster coordinated movement and develop weapons handling skills than trying to build strength.
In terms of nutrition, you have to draw a distinction between rations in the barracks/fortresses and what was eaten on campaign, but I'm not really qualified to speak to that.
As for nutrition, Philip K. Roth's Logistics of the Roman Army at War was very detailed in describing the diet of a Roman legionary on campaign. Each soldier would receive rations of wheat (bread or biscuits), oil, vegetables, meat (usually salted) vinegar, cheese, and wine (or potentially beer and butter, on the northern frontier). This diet provided at minimum 3,000 calories per day, including plenty of protein for muscle building, and was sufficiently varied to prevent malnutrition of micronutrients as well.
In camp the diet would be more varied based on local foodstuffs, although generally the Roman camp diet influenced the local diet as much if not more as the reverse. They still relied on their rations, which may well have come from several far-off parts of the empire to provide the bulk of what they ate.
Training was another matter, the Legions drilled frequently, engaged in marches even when not on campaign, and completed building projects- an often strenuous activity to be sure. As another post states, endurance was more important than raw strength, as should be illustrated by the roman fighting style. The gladius was primarily a stabbing weapon, and thus did not require the strength necessary to wield a long sword or an axe, which were more often used for powerful (but tiring) overhead chops.
The classic soldier of antiquity was built for efficiency, so muscle was only important in certain key areas. One of the most frequently-cited descriptions comes from Vegetius in De Re Militari from the 4th century CE:
"Let, therefore, the youth who is to be chosen for martial tasks have observant eyes, hold his head up, have a broad chest, muscular shoulders, strong arms, long fingers, not too extended a waist measure, lean hams, and calves and feet not distended with superfluous flesh but hard and knotted with muscles. Whenever you find these marks in the recruit, do not be troubled about his height. It is more useful for soldiers to be strong and brave than big."
Notice where the muscle is supposed to be: in the calves, not the thighs, shoulders and arms, not the chest. This is about efficiency and endurance. The larger the soldier, and the more massive his long muscles, the more food he must consume to maintain his musculature. Starvation and limited food were simple realities of ancient warfare. Suetonius famously comments on how Caesar's soldiers, while blockading Pompey in 48 BCE, were forced to eat bread made from grass just to fill their stomachs. According to his account, they catapulted the bread into Pompey's camp just to show how tough they were and how little they needed to survive on, and how Pompey had no chance against them. Pompey supposedly hid the bread from his men, saying "These are not men, but wild animals I fight!"
In other words, your ancient soldier was lean and efficient, with no extraneous muscle, and muscle was built by the marching, digging, and fighting they did, with little energy wasted while in the field. Being able to survive, work, and fight on low rations was essential to success.
Research on 19th century Zululand (in what is now South Africa) suggests that stick fighting was a complex activity embedded in the daily life of Zulu boys. It was about so much more than just training their bodies for warfare, though: “stick fighting long adhered to rules of competition that privileged rhetoric, honour and defence.” So you didn't just pick up a stick and smack your opponent. It was a respected sport, an activity that you did within the confines of well understood rules, and with some kind of supervision, too.
Moreover, the stick itself had significant symbolic meaning. "[As] Ndukwana kaMbengwana, an oral historian of the Zulu kingdom pointed out in 1903, a boy who received his original stick knew he held more than a weapon or switch. His stick epitomised a customary obligation to shield his lineage resources from any harm[...]." So while you were learning about fighting, you'd be doing so in the context of protecting the family's cattle, its source of wellbeing.
So all the rituals and behaviors and traditions of stick fighting helped train young men’s minds, helped teach them about what it meant to be an honorable, well-spoken man. Stick fighting “imbued Zulu masculinity with an ethos of self-control” that was important for more than just military discipline—that kind of mental strength, researchers argue, was very important for surviving the upheaval brought about by colonialism.
The article I’m referring to is Carton, B., & Morrell, R. (2012). Zulu Masculinities, Warrior Culture and Stick Fighting: Reassessing Male Violence and Virtue in South Africa. Journal Of Southern African Studies, 38(1), 31-53.