You would still have the same age, gender, personality, skin color, etc. and you would be able to speak at least one local language and would know basic information of the era and place. Your family, social standing, and such would be randomly picked.
Shitty ultraconservative social structure and mass serfdom but looking at their history they usually ended up worse after even victorious war - due to their shitty social structure rather than war itself. Though it still took them over a century and something along 5 lost wars in a row to end up in the dustbin of history.
Today Sparta is a small town mostly unrelated to the ancient one except rough location, it was depopulated for centuries and refounded by king Otto in 1834. So it sounds weird they don’t consider themselves Greeks, why would they since they literally are. Anyway they don’t have much to do with ancient Spartans since ancient Sparta has been totally razed and its population sold into slavery by Visigoths, and later Slavs raided the area multiple times in VI and VII century, causing remaining population to flee.