The way God judges things is very different from how humans judge. In judgment humans point to the wrong you did in convicting you of your offenses. But God points to all the right you didn't do hence convicting you of your sins. In judgment God points the world to Christ and by the time they see Christ they become convicted of there sins. This is what Jesus meant when he said "And when He [the Holy Spirit] has come, He will convict the world of sin, and of righteousness, and of judgment:" (John 16:.

The verse is saying the world is going to be judged by righteousness and the result is conviction of sin. How is God going to use His Spirit to reveal righteousness to the world? Jesus answer this when he said "because I go to My Father... He [the Holy Spirit] will glorify Me, for He will take of what is Mine and declare it to you. " (John 16:10 & 14). Jesus is saying the 'Holy Spirit with reveal God's righteousness to the world by taking my righteousness and declare it to you so that the world would sees my righteousness in you and be convicted of their sin.' This is how God is going to judgment the world by His Spirit.

Note: By "judgment" I am talking about the convicting aspect of judgment and not the executing aspect. In judging someone you have to convict the person before executing the necessary punishment. It is the convicting part of judgment I am referring to.

God also judges us in the same way. God doesn't convict us by pointing us to our sins, it is the devil that does that. God convict us by pointing us to Jesus and by the time we see his righteousness and purity we are convince of how sinful we are.
Man can never realize how sinful they are until they see Christ in his purity. The more of Christ we see the more we realize the depth of our sinfulness and then we desire more of Christ cause we now see that "in me (that is, in my flesh) nothing good dwells;" (Rom 7;18 ). It is clear in the Bible that on judgment day God is going to judge the world through Christ, but how is He going to do that? Is He going to start showing everybody all the sins they have committed? No, God is going to reveal Christ in his righteousness and purity and when the world sees him each person would realize how sinful they are.
Yes by the time they see true righteousness they would realize all their unrighteousness, self-righteousness or false righteousness. This is how God would judge the world because this is God's way of judgment, God's way of judgment is always Christ.

Yes Christ is both the Judge and the Judgment of God.