In this synoptic history of the early Nephites, Nephi appears to be relating events in a rough chronological order (or at least the ordering is suggestive of a chronology). After basing his earliest efforts on their religious beliefs, they concentrate on the most important things - food and protection. With those elements secured, and with a larger (and now more skilled) population, Nephi can now turn to the construction of a significant sacred structure - a temple.
The very building of such an edifice suggests a substantially increased population. While we don't know how long after their arrival in the land of Nephi they began to build this temple, the very act of public building requires a fairly large base population. It requires that there be sufficient stability of food sources to be able to spare labor for the public building. It requires that there be sufficient surplus that the basic needs of the people are covered and there is yet material and effort available for the public structure. All of these are present.
We understand that they were not yet present in abundance, however, because Nephi clearly states that this temple, while modeled after that of Solomon, is "not built of so many precious things." While there is surplus sufficient to build, there is not sufficient to supply the truly luxury items that would be associated with the temple of a reasonably prosperous people such as Solomon governed.
We do not know the material of which the temple was built. The early date of the building would suggest that it might have been wood rather than stone. Stone would have required learning a new set of skills not specifically known to have been in Nephi's repertoire, and would also take much more time and labor.
In Mesoamerican, temple building was a very old tradition, and while this temple was modeled after that of Solomon, it does not take much imagination to transfer to a different mode of temple-building once a shift was made from wood to stone. When stone began to be used, the methods and models for stone construction would come from the surrounding cultures, and the resultant temples would likely have begun to look like the temples of the area. The importance of a temple is its sacred nature, not its construction.