I've read that the popular Roman characterization of Persia, Egypt, and to a lesser extent Greece was that people in these places were decadent and effeminate, compared to the Romans who valued more masculine, stoic virtues. But I can't imagine that men in Persia and Egypt would have characterized themselves this way, so what type of person did the men of the ancient Near East strive to be, and did they associate virtue with masculinity? The two examples of "manly men" that come to mind are Gilgamesh (millennia removed from the Persians/Romans) and Samson from Judges (a great warrior, but ultimately not a model to emulate). Thanks for any clarity you can provide and sorry if my perception is totally off-base.
The answer to your question will depend on which period you are interested in. If you are talking about the time of Gilgamesh, the Codex Hammurabi will be a good vantage point. These are the first laws that were written down (to have been discovered.) Of course laws are meant to be objective, but in trying to be objective they reflect the normative values of the society which created them. I haven't actually gone through and analysed these myself, but you should quite easily be able to find secondary literature where such analysis has been done. The references on the Wikipedia article might be a good starting point.
In later Babylonian time, Amélie Kurth provides this anecdote of their society which I find illuminating.
[...] an important ritual during the New Year festival in Babylon consisted of the king being stripped of his royal insignia, struck in the face, forced to his knees before Marduk [patron god of the babylonian pantheon and tutelar deity of Babylon] by having his ears pulled, and then making this solemn avowal:
I have not sinned, lord of the lands,
I have not neglected your godhead,
I have not destroyed Babylon,
I have not ordered her to be dispersed,
I have not made Esagila (main temple in Babylon) tremble
I have not struck the people of the kiddinu [assumed to be citizens or a privileged group of the Babylonian population] in the face
I have not humiliated them.
I have paid attention to Babylon,
I have not destroyed her walls.
See Kurth (2014) Even a dog in Babylon is free. I got the quote from her book The Ancient Near East 3000-330 BC.
Then in the Greek golden age, there's the story from Herodotus that the Persians learned of love of boys from the Greek, something his (more or less) contemporaries would disagree with. Wikipedia has some nice info. This seems like effeminate behaviour by our standards, but was not necessarily that at the time.
One of the main problems of the perception of the ancient Near East is that it mainly comes from a Greek vantage point. It's a major problem of the field of Assyriology that there aren't any writers like Plutarch, Aristotle or Herodotus who wrote major works of philosophy of good government or history. Therefore it's a about piecing together circumstantial evidence into a narrative. I'm by no means an expert on the field, but I would recommend to look into something like the article I linked by Kurht or even the whole book. Another book on the subject that I've enjoyed was L. Oppenheim (Chicago UP 1977) Ancient Mesopotamia - Portrait of a dead civilisation.
Hope this helps.
Edit: fixed links
The best way to understand such a thing would be through it's primary sources. Literature like the Epic of Gilgamesh, the Jewish, Christian and Muslim canon, Persian works like Shahmasheh. Although these come after, from time to time, such as the beginning of Genesis, it explores those who came before them. The Jewish series of books about its Judges - that is, military leaders, from time to time will describe the ancient culture of the Philistines. Of course, every source must be taken from the specific grade of salt particular to that culture. So as a historian, the key is to appreciate the many recordings of the ancient near east, as described by their peers. For direct evidence, archaeology is the best we have to offer.