A number of theories have been proposed, as I have been very interested in this topic since Solkar has become such an important character in my YMAM series. Ideas include that he was on the ship but not the captain; that he was a diplomatic envoy present on a scout ship *just in case* they made a first contact so it could be done properly; that he was not a diplomat but was on the ship or was captain and simply became a diplomat later because of his unique experience and relationship with Cochrane and early Human-Vulcan relations; and that he just inherited the ambassadorship even though he wasn't really the Vulcan who stepped off the ship.
Well, without further ado, for your consideration and critique, I present some new evidence on MEmory Alpha:
According to the Star Trek Customizable Card Game, Solkar was the Vulcan captain of the T'Plana-Hath who greeted Zefram Cochrane in Star Trek: First Contact.
In an unpublished interview with Larry Nemecek, "The Catwalk" writer Mike Sussman believes this as well, citing the wording in Archer's dialog in "Horizon" in which he refers to Cochrane's first contact and says, loosely, "I don't remember anyone going after the Vulcan ambassador with pitchforks and torches when they first landed."