Acts 8:13 tells us that Simon believed Philip’s preaching concerning the kingdom of God and the name of Jesus Christ; however, in 8:21, Peter tells Simon that his heart is not right in the sight of God. Is Simon a true believer? Because it would be simple if v. 13 read, “Simon said he believed,” but it doesn't. And what does v. 23 mean when Peter tells Simon he is in the gall of bitterness, and in the bond of iniquity.
The simple answer to your question is “no” - Simon is not a true believer. It is helpful when reading the book of Acts to keep in mind that it is a transitional book, from Judaism to Christianity. We do not look for doctrine in this book. At the beginning of Acts the Holy Spirit was received by the laying on of the hands of the apostles (8:17), whereas today He indwells the believer upon salvation. The people of Samaria believed when Philip preached the glad tidings concerning the kingdom of God and the name of Jesus Christ and were baptized. Baptism is the way of entrance into the sphere of Christian profession; the possession of the Holy Spirit is the evidence of new life. Simon does not have the Spirit and so is not saved - does not have eternal life. So his belief, like all “professers” is intellectual and without reality. There are many who believe with the head, but not the heart (Rom.10:9); the demons also believe (Jam. 2:19). During the ministry of the Lord Jesus Himself we know that many believed, but He did not commit Himself to them (Jn. 2:22-23). Peter’s response makes clear that Simon could repent, but that he was still in bondage and money was controlling his life. Simon is looking for a man to intercede for him before God, rather than turning to the Lord Jesus, who alone gives faith for belief.