What did Jesus' death on the cross accomplish if everyone is to be judged on judgment day?

There are two simple things to be aware of in order to answer this question. First, by what standard, does God judge a person, and second, what are the judgements of God that will ultimately involve everyone born into the human race. Let's have a look at the details.

By what standard, does God judge a person

The anticipation of a coming Redeemer (and Messiah, for Israel), the subsequent birth and life of Jesus Christ, and (for us in the grace dispensation,), the death, burial and resurrection of Jesus Christ, is the absolute standard and singular salvific event of God's great redemption plan for humanity. The pivotal point in all of time rests on the life and sacrifice of Christ. Ultimately, every individual person will be judged by the faith they place in God's Son, Jesus Christ.

Gal.4:4 But when the fulness of the time was come, God sent forth his Son, ...

What are the judgements of God

It is true that everybody will be judged. Hebrews says it as follows,

Heb.9:27 And as it is appointed unto men once to die, but after this the judgment:

However, not all people will be judged to the same degree, nor at the same time, or at the same judgement event. There are several judgements that God performs throughout His plan. To have a better understanding of these judgements, you can read up on them in this post, but as an answer to this question, I'll refer to only two, the Bema Seat and the Great White Throne judgements. Comparing these two will make it clear that the life, and death, of Christ, certainly accomplished much and changes the fate of many!

God's salvation plan extends across the entire history of humanity. With this in mind, God provided people in different times of history with different instructions, or gospels, (which is good news), for salvation. Essentially, people are justified by faith in a gospel that God has revealed in their own time frame.

It's both the life and the death of Christ that saves us from judgement

So, for people who lived from Adam to Christ, thus, before Jesus' death and resurrection, they had to trust in the Lord and believe in a coming Redeemer whom God would provide as a perfect sacrifice to free them from sin and save their souls, —a trust they showed in offering sacrifices. That believer, by sacrificing an animal, was saying, "I trust God for a future Redeemer (sacrifice) that will save me despite the fact that I am sinful", as God has taught before, Gen.3:21. This was the means, during that time, to stay in relationship with God and avoid his judgement.

Genesis 3:15 And I will put enmity between you [Satan] and the woman, and between your seed and her Seed; He shall bruise [crush] your head, and you shall bruise [strike] His heel. ... 21 Also for Adam and his wife the Lord God made tunics of skin, and clothed them.

Hebrews 11:4 "By faith Abel offered to God a better sacrifice [a blood sacrifice] than Cain, through which he obtained the testimony that he was righteous, ...".

Then, for people who lived after Christ's death, which includes you and me today, they had to believe in the death and resurrection of Christ, which was the work of God through Christ and becomes the gospel that was now to be received and believed.

Ephesians 2:8-9 For by grace you have been saved through faith, and that not of yourselves; it is the gift of God, 9 not of works, lest anyone should boast.

Romans 11:6 And if by grace, then it is no longer of works; otherwise grace is no longer grace. But if it is of works, it is no longer grace; otherwise work is no longer work.

Philippians 3:8 Yet indeed I also count all things loss for the excellence of the knowledge of Christ Jesus my Lord, for whom I have suffered the loss of all things, and count them as rubbish, that I may gain Christ 9 and be found in Him, not having my own righteousness, which is from the law, but that which is through faith in Christ, the righteousness which is from God by faith;

What did Jesus' life and death ultimately accomplish?

To wrap up this answer, we must understand the facts. Everyone will be judged, but those that believed God's good news of a coming Redeemer, or God's good news that the cross-work of Christ is payment for their sins, it is these people who will be saved from the wrath of God in His righteous judgements.

Both the life of Christ, and His death and resurrection, accomplished salvation for people. It is there to be received as a free gift. For all those who believed in the good news of Christ, whether they lived before or after His death, these will be judged as innocent and will live in the sight of God. But for those who rejected the good news of Christ in their lifetime, these will be judged as guilty and will suffer God's wrath and be subject to eternal death. 

What did Jesus' death on the cross accomplish if everyone is to be judged on judgment day? It accomplished much! It accomplished life and redemption, but only for those who choose to believe in Him.