I'm sorry for such an open question, but I've come up against such a total lack of information that ANYTHING would be really useful. I'm wondering especially about passing and/or transitioning, if that was even possible.
Elagabalus, for example, apparently attempted to have gender affirming surgery (even though it's possible that was a rumour, it still indicates such surgery's unavailability). But that's a completely different society, with different technology and medicine and social constructs. I'm wondering if anything like that exists for the Ancient Near East. I'm especially interested in Assyria, but I'll take what I can get. Thank you!
Apologies for the delayed response! Anyhow, if you still need research materials on the topic Gwendolyn Leick's Sex and Eroticism in Mesopotamian Literature is a good starting point and it has a discussion of several figures and functionaries that appear in texts to take on multiple gender or gender-ambiguous roles, most notably the Asinnu. One caution I would make is that most of the figures discussed are some kind of cultic functionary or set role. So we don't know that there were trans* individuals as we use the terms today(that is, people who shifted from life as an ordinary man to life as an ordinary woman or vice-versa) and surgery would not have been possible, but there were perhaps social roles that occupied a liminal gender and which could be taken on by people whose internal sense of gender did not match their external genitalia.