A non-Christian friend of mine says that Jesus was a false prophet because he wanted to eliminate almost all the Jewish customs and laws. Is this true?

No, Jesus was and is a true prophet. Jesus says he has not come to abolish the law and the prophets but to fulfill them (Matthew 5:17). Moses prophesied that a prophet like himself would one day be raised up (Deuteronomy 18:15-19) and, following the Resurrection, both Peter (Acts 3:22) and Stephen (Acts 7:37) testify that Jesus fulfilled Moses’ prophecy.
 
A key way Jesus demonstrated that he was a true prophet was in telling the Pharisees that no sign would be given them except the sign of Jonah (Matthew 12:38-40). He demonstrated this first by rising from the dead, as Jonah was three days in the belly of a whale (Jonah 2:1). But the other aspect of the sign of Jonah was getting the Gentiles to repent and turn to God. For Jonah, it was getting the people of the great city of Nineveh to repent.
 
With Jesus, we see something much more dramatic. Jesus commissioned his apostles to make disciples of “all nations,” including the “Gentiles” (Matthew 28:19). We see that accomplished in a significant way through the Catholic Church he founded. Catholic means universal, which is appropriate, since with God’s fundamental support the Church has made disciples throughout the world.
 
Jesus is the Incarnate Word, the God-man (John 1:1-3,14). Jesus cannot help but be a true prophet and indeed the greatest ever.