Just like justification by faith, the act of God providing us his righteousness (also known as "imputed righteousness") isn't really a new concept. David touched upon this concept when describing "the blessedness" (4:6).
In particular, David knows that the people "whose lawless deeds are forgiven" and the people "whose sins are covered" are blessed (4:7). Additionally, the people "to whom the Lord shall not impute sin" are also blessed (4:8). These two quotes are from Psalm 32, which were written after David realized he made a major sin mistake (which had some natural side effects like guilt). In response to realization of sin, David turned to God for forgiveness, and after receiving said forgiveness, David feels blessed and describes the blessing of God's forgiveness.
Just like David, we who accept Jesus's righteous gift are also blessed because of how Jesus provided his righteous Spirit. Although there are still natural side-effects of sin, we are forgiven by God, meaning that our sins will ultimately not separate us from God's place in the end. And because of God's forgiveness, we can get back up.