While it's true that STAR TREK is full of contradictions and discontinuity (and specific history about the Valiant does seem vague), we can establish that nobody (from Earth) ever launched or travelled in an FTL vehicle prior to Zephrem Cochrane in April of 2063. While "Metamorphosis" and STAR TREK: FIRST CONTACT don't match up very well, they are close enough to make it clear that Cochrane did what nobody from Earth had ever done before: he discovered the space-warp and built and piloted the first warp-propelled Earth spacecraft.
The 2018 reference in "Space Seed" does dangle there, since it seems obvious that any early-21st century Earth vessel predating Cochrane's Phoenix flight would have to be sublight in nature. Of course, FIRST CONTACT seems to have "moved the goalposts" by shifting the third world war from the 1990s to the 2050s, so it seems logical that Bermanian revisionism/retconning in FIRST CONTACT re-establishes Lt. MacGyvers' "2018" sleeper ship reference down the road 55 years to 2063.
Since the S.S. Valiant was obviously one of the first Earth warp-propelled spacecraft to be launched into deep space, and since it should be obvious that Earth did not have FTL capability until Cochrane's Phoenix flew in April of 2063, it should be pretty safe to say that the Valiant was built and launched some time in the 2063-2065 time range, definitely after the Phoenix. We could give ourselves a little wiggle room since mentions of the Valiant in "Where No Man Has Gone Before" seem to be wishy-washy on the exact number of years; Kirk's log entry seems to place the Valiant as being from "over two centuries" but subsequent mentions in the story aren't quite as clear. Didn't Kirk say subsequently that the recorder-marker had been ejected "almost two hundred years ago"? That would leave a little bit of wiggle room there. Not much, but a little.