JOEL - The way we understand this is that the Nephites did not hold the lesser priesthoods, namely the Aaronic or Levitical priesthoods, because they were not descendents of Aaron or Levi. But all prophets from the beginning of time(including Lehi) had the higher or Melchizedek priesthood (Gen. 14:18, Ps. 110:4).
The Melchizedek Priesthood was to have continued among the descendants of Israel:

"Now therefore, if ye will obey my voice indeed, and keep my covenant, then ye shall be a peculiar treasure unto me above all people: for all the earth is mine: And ye shall be unto me a kingdom of priests, and an holy nation. These are the words which thou shalt speak unto the children of Israel." (Ex 19:5-6)

However, because the children of Israel were disobedient in the days of Moses, the keys of the Melchizedek Priesthood were withdrawn from them.(D&C 84:23-25) (see also Joseph Fielding Smith, Doctrines of Salvation, 3 vols., Bruce R. McConkie, The Mortal Messiah, 4 vols., Salt Lake City: Deseret Book Co., 1979-81, 1:60).

But even though the children of Israel after the time of Moses could not hold the Melchizedek priesthood, all the succeding major prophets were still allowed to hold it so they could lead with authority from God.
The Book of Mormon does not specifically state how Lehi got it, but according to Joseph Smith:

"All the prophets had the Melchizedek Priesthood and were ordained by God himself". (See Teachings of the Prophet Joseph Smith, pp. 180-81).

This would include Lehi. And it is obvious the Nephites certainly knew about the Melchizedek Priesthood as evident in the prophet Alma's detailed description of it.(See Alma 13).

It is possibile that the Priesthood was passed on from father to son. The prophet Joseph, who was sold into Egypt, may have passed the Melchizedek priesthood to his son and so on until Lehi received it from his father.

"And it came to pass that my father, Lehi, also found upon the plates of brass a genealogy of his fathers; wherefore he knew that he was a descendant of Joseph; yea, even that Joseph who was the son of Jacob, who was sold into Egypt, and who was preserved by the hand of the Lord, that he might preserve his father, Jacob, and all his household from perishing with famine. (1 Nephi 5:14)

Lehi may have even received the Melchizedek Priesthood from his Old Testament contemporary prophet, Jeremiah, while he was living in Jerusalem.

Although baptism is not mentioned in the Old Testament we know it was done before Christ came, by people like John the Baptist(John 1), he being a direct descendant of Aaron.
The practice existed as far back as the time of Adam.(Moses 6:64-66), and has existed on the earth as long as there was priesthood authority to perform it (D&C 84:27-28).