Should I repeat the Fajr prayer if I dozed off slightly during the first prostration and did not wake up until the second prostration of the first rak'ah, meaning I only performed one prostration in the first rak'ah, and is there any expiation due, given that this happened a long time ago?

If you were not an adult when that happened, then there is nothing upon you, and you must take care of your prayers in the future, due to the saying of the Prophet, peace and blessings be upon him: "The pen has been lifted from three: from the sleeper until he awakens, from the imbecile - or he said, the madman - until he regains his sanity, and from the child until he reaches puberty."

However, if you were an adult, then your prayer is invalid and you must repeat it, either due to omitting the prostration, because prostration is obligatory by consensus and by Sunnah, or due to heavy sleep which nullifies ablution, due to the saying of the Prophet, peace and blessings be upon him: "The eye is the strap of the anus; so whoever sleeps, let him perform ablution."