"By faith Noah, being warned of God of things not seen as yet, moved with fear, prepared an ark to the saving of his house; by the which he condemned the world, and became heir of the righteousness which is by faith" (Hebrews 11:7).
Noah's obedience serves as an example for us for several reasons. First, Noah's obedience required that he make a start. The 120 years in Genesis 6:3 is most likely a reference to a one hundred and twenty year period that God allowed to expire before sending the flood, thus providing man with an opportunity to repent (II Peter 3:9). Noah could have procrastinated, but did not. Our obedience requires that we make a start. Unfortunately, many people procrastinate for a number of "reasons" (excuses? - too young, don't know enough, pressured not to by my parents, etc.).
Second, Noah submitted to God's plan. God gave Noah a specific plan whereby Noah and his family could be saved from the flood (Genesis 6:14-16). God has given us a specific plan whereby we can be saved from our sin (Romans 10:17, Mark 16:16, Acts 17:30, Romans 10:9, Acts 2:38, Romans 12:1-2). Noah did not have the freedom to alter God's plan, nor do we. We must submit to all God has commanded (Genesis 6:22, Genesis 7:5, John 14:15, John 15:14). We often emphasize the necessity of confession and baptism but trivialize other commandments (ex: controlling our anger - Matthew 5:22, Colossians 3:8, modest dress - I Timothy 2:9, pure speech - Ephesians 4:29, Colossians 3:8, keeping our word - Matthew 5:33-37, and fairness in business matters - Romans 12:17).
Third, Noah shared God's plan. The scriptures teach Noah was a preacher of righteousness (I Peter 3:19-20). We have a responsibility to take the gospel to the lost. Consider Matthew 5:13-16, Matthew 28:18-20, along with II Timothy 2:2.
Fourth, Noah stuck to God's plan. Because of Noah's steadfastness, God "shut him in the ark" along with his family (Genesis 7:16). Christians are to remain steadfast (I Corinthians 15:58) and refuse to give up (Galatians 6:9).