Noah did everything just as God commanded him.
Again we see a command that is meant to bring life. Certainly if Noah hadn't obeyed God it would have brought death, a stark contrast to Adam's response to God's command.
Roman's 1:5 speaks of an obedience that comes from faith. God wants our obedience to be because we love and trust Him not because we feel we have to or have no choice in the matter.
There is an obedience that comes from faith - Noah had faith and obeyed God. Many others believed God's promises and were obedient when God asked something of them and their obedience often brought great rewards especially a closer relationship with their Heavenly Father.
All who believe today have an advantage in that God, through His New Covenant brought about by Yahshua has written His commands on our hearts and minds so that we might delight in doing His will.
It is His Spirit that brings about our obedience through faith.