Do Book of Mormon prophets similarly serve as "types" of Christ? I believe they do, at least those who lived before the coming of Christ. Elder Bruce R. McConkie said, "A prophet is one who has the testimony of Jesus, who knows by the revelations of the Holy Ghost to his soul that Jesus Christ is the Son of God. In addition to this divine knowledge, many of them lived in special situations or did particular things that singled them out as types and patterns and shadows of that which was to be in the life of him who is our Lord."  Elder McConkie said nothing to differentiate between prophets in the Old Testament and those in the Book of Mormon.
My first example of such a parallel is the last Jaredite prophet, Ether, who bore witness to the destruction of his entire civilization. In addition to bearing the mantle of his nation's prophet, he was also the rightful heir to its throne; Ether was a descendant of Orihah, the first Jaredite king (see Ether 1:6-32), but Ether's grandfather, Moron was dethroned and "dwelt in captivity all the remainder of his days," as did Ether's father, Coriantor (see Ether 12:17-23).
Similarly, Jesus Christ, in addition to bearing the mantle of Israel's Messiah, was also the rightful heir to the earthly Davidic throne. As Elder James E Talmage noted, "Had Judah been a free and independent nation, ruled by her rightful sovereign, Joseph the carpenter would have been her crowned king; and his lawful successor to the throne would have been Jesus of Nazareth, the King of the Jews." Talmage further observed that, among all the charges of blasphemy leveled against Him by the Sadducees and Pharisees—folks who knew a little about Hebrew genealogy—not once did they say He was ineligible to his claim of Messiahship because of lineage .
Ever wonder why we're bothered with Ether's genealogy in Ether 1? Perhaps, in part, it's to further emphasize this parallel, as similarly extensive genealogies of the Savior are given in Matthew 1 and Luke 3. Regardless, neither were very concerned about these earthly inheritances, as both devoted their attention to a kingdom which was "not of this world" (John 18:36) but of a "better world" (Ether 12:4).
 see Andrew Skinner, "Finding Jesus Christ in the Old Testament." Ensign June 2002. <http://www.lds.org/ensign/2002/06/finding-jesus-christ-in-the-old-testament>
 Ibid or whatever.
 Bruce R McConkie, The Promised Messiah. p. 448
 James E. Talmage, Jesus the Christ. p. 87