In John 16:7, Jesus promises the coming of the Holy Spirit, which is fulfilled in Acts 2:33 when Peter explains that Jesus, exalted to the right hand of God, has poured out the Holy Spirit.
John 16:13 speaks of the Spirit of truth guiding believers into all truth, which is echoed in 1 Corinthians 2:10-13 where Paul describes the Spirit revealing the deep things of God.
John 16:20 contrasts the disciples' sorrow with future joy, similar to Psalm 30:5, which speaks of weeping enduring for a night but joy coming in the morning.
John 16:33 assures believers of peace and victory over the world, which is reinforced in Romans 8:37 where Paul declares that believers are more than conquerors through Christ.