The exodus from Egypt was in the year 2448 from the creation of the world (Rashi A”Z 9a; Seder Olam Zuta chapter 1). Melachim 1:6:1, “And it was in the 480th year of the leaving of the children of Israel from the land of Egypt, in the fourth year in the month of Ziv, which is the second month, of Solomon being king over Israel, and he built the house for HaShem.” Rashi (A”Z 9a) says that this was in the year 2928.
But the 480th year from leaving Egypt is 2927. This counts the first year (2448), as Shemot 40:17, Bamidbar 1:1, Bamidbar 9:1, and Bamidbar 10:11 count the year after the exodus as the second year. Although the years from leaving Egypt are counted from Nissan to Nissan (Rosh Hashana 2b and 3a), the 480th year from leaving Egypt was from Nissan 2927 through Adar 2928. Since the Beit Hamikdash was built in the second month of Iyar (before Tishrei 2928), the year was still 2927.
Furthermore, Jewish king years also count from Nissan to Nissan (R”H 2a), and the 4th year of Solomon aligns with the 480th year from leaving Egypt including the first year of 2448. Zevachim 118b-119a counts the 480 years as follows: 40 years of the desert, 39 of which had the Ohel Moed of the desert that was built on the first day of the second year, 14 years of the Ohel Moed in Gilgal when they conquered and divided the land, 369 years of Shiloh, 13 years of Samuel and Shaul, 40 years of David, and 4 years of Solomon.
King years also count the first year as first. See Megilah 11b where “in the third year of his reign…” (Esther 1:3) of Achashveirosh refers to his third year, and “in the third year of the reign of Belshatzar” (Daniel 8:1) refers to his third year. Furthermore, see Rosh Hashana 2a that if a Jewish king was appointed on the last day of Adar, once Nissan arrives, it counts for a full year. Rashi explains that the king now starts his second year. Since Melachim 1:6:1 and DH 2:3:2 state that the building of the Beit Hamikdash was in Solomon’s 4th year and since the years of kings also count the first year as the first, this further shows that the first Beit Hamikdash was built in the year 2927.
The 40 year decree of the desert includes the first year (2448). This is why the Ohel Moed of the desert lasted 39 years. Furthermore, Shemot 16:35 states that the children of Israel ate the mon for 40 years. The mon fell on the 16th of Iyar (2488) (Shemot 16) and lasted until the 15th of Nissan (2488) when they entered the land (Joshua 5:12). Kedushin 38a and Seder Olam (chapter 15) further explain that they tasted the taste of mon in the cakes they took out of Egypt. The mon frames 40 years from leaving Egypt (Nissan 2448) until entering the land (Nissan 2488).
Furthermore, Seder Olam (chapter 15) states that the first Beit Hamikdash was built at the half-point of the jubilee in the 4th year of shemita (according to the Rabbis who use 50 year cycles) in the 4th year of Solomon. The first ever year of shemita was 2503 (Rambam Hilchos shemita chapter 10 halacha 2). Consequentially, 2902 was a jubilee, and 2927 was the 25th year of the jubilee (half of 50) and the fourth year of shemita.
Zevachim (above) counts 57 years for the Ohel Moed in Nov and Givon after Shiloh, which include 13 years of Samuel and Shaul, 40 years of David, and 4 years of Solomon. Rashi there learns that Solomon ruled for 4 years before building the Beit Hamikdash. Rashi learns “in the fourth year” of Solomon to mean after the fourth year. Rashi (A”Z 9a) also learns “in the 480th year” of leaving Egypt to mean after the 480th year. Rashi will also learn that the first Beit Hamikdash was built after the 25th year of the jubilee.
The simple approach is that the first Beit Hamikdash was built in 2829 (in the 480th year of leaving Egypt, in the fourth year of Solomon, in the 25th year of the jubilee). Since the first Beit Hamikdash started in Iyar, the 4th year of Solomon (2927) contains both the 57th year of Nov and Givon and the first year of the first Beit Hamikdash.