If grace and truth came only through Jesus Christ, doesn't it mean that grace was there from the beginning?

No, it wasn't. God existed from the beginning, but no human being got grace because it was not possible until man's sin was forgiven and the Holy Spirit came in.

The Holy Spirit is called in Hebrews 10:29 "the Spirit of Grace." The Holy Spirit could not come into a man's heart until that heart was cleansed from sin, and nobody's heart was cleansed from sin until the day of Pentecost. Before that they only had an external offering of a sacrifice to look forward to the death of Christ. But, when the heart was cleansed, then the Holy Spirit could come. The blood of Jesus had to cleanse our heart first, and it's only happened from the day of Pentecost onwards.

So, that's the time the Spirit of Grace brings grace. Grace is not some thing that God gives us. It's the Holy Spirit coming in and giving it to us, and it's only then that we experience grace that helps us to overcome sin. Sin will not have dominion over you, because you're under grace.

The first person to walk on this earth who was under grace was Jesus Christ. Luke 2:40 says as he grew up, "the grace of God was upon him," and therefore he never sinned.