Solomon is, of course, the writer (as we will see in a moment) and it is certainly something that is applicable for himself as he will be the shepherd that the Shulamite woman falls in love with. Solomon can do nothing but write about himself, seeing as this is what he was experiencing. The important thing to note and remember through this reading is (whether Solomon understood it or not) this wasn't about him. It was about Christ. All of scripture was/is about Christ.
There are a plethra of books writen and recorded by Solomon that are not even located in scripture. Some are found in books like the Apocrypha, and others are books that have not made their way into scripture (perhaps things like journal entries). Other holy books even talk about Solomon and his glory (including, but not limited to the Quran).
Solomon is a widely known figure in the religious circles, but there was something that shouldn't lost on mere religion alone. Solomon knew the God of the Bible. He conversed with God. He desired to lead the people FOR God. Jerusalem needed a Godly leader, and Solomon was the one chosen for the job. Through all his wisdom, he eventually understands the relationship aspect that God desires, and I believe that God reveals this through this story/song.