The Bible says that there were many saints who ascended to be with God before Jesus died (for some examples, see Genesis 5:24, 2 Kings 2:11, Hebrews 11, and Luke 16:22). However, the Bible is not abundantly clear as to whether they ascended to heaven or simply a temporary place of rest. Adding to the mystery is how the Bible talks about Sheol or Hades as a common place for all dead, both believers and unbelievers.
Regardless of the mystery of those who died before Jesus came, Jesus was not saying that He would go to heaven before others in John 3:13. Instead, He was teaching Nicodemus that His message was authoritative because it came from heaven. He was also saying that, as the One who ascends and descends to heaven, He is God (c.f., Proverbs 30:4). We cannot answer the question either way about whether anyone went to heaven before Jesus’ ascension using John 3:13 because that is not what Jesus intended to teach. Instead, we need to understand what Jesus was saying, which is that salvation comes through Him, the Son of God (John 3:16).
Many people died and were in the presence of the Lord before Jesus ascended to heaven (Hebrews 11). That means that they, at the very least, went to a place of rest. Were they in heaven, proper, before Jesus completed His work? That is unclear. However, what Jesus was talking about in John 3:13 was that His message of salvation was authoritative. He was also implying that He was more than the messenger—He was God, since only God ascends and descends from heaven.
Since Jesus’ message is divinely authoritative, it is important that we seek to understand it so we respond rightly to it. The message was summed up in Jesus: “so that whoever believes in Him may have eternal life” (John 3:15). Therefore, we are called to believe in Him. Jesus is God’s “only Son” (John 3:16), which means that Jesus is also God. Jesus came to die as a propitiation—a substitute who dies for our sins—for all men who believe in Him (1 John 4:10). So, believing in Jesus means that He is the Savior from God’s wrath (John 3:36; c.f., Romans 5:9). God’s wrath is coming because “people loved darkness” (John 3:19). Indeed, all of us love darkness (Romans 3:9–18) and are deserving of the coming judgment. However, “the Light (Jesus) has come into the world” (John 3:19b). When we believe in Him as God’s Son, Savior, and a sacrifice that turns away the wrath of God (1 John 4:10), then we believe His message and are given salvation!