What is the Bible verse on how the the anticipation of an evil is often more fearsome than the evil itself?
The prophet Amos spoke of a time when there was such evil oppression, bribery and corruption of justice that "Therefore the prudent man keeps quiet in such times, for the times are evil" - Amos 5:12-13 N.I.V. Anticipation of how badly it will fare for wise men at the hands of evil men cause the prudent to keep quiet. Yet that is not spoken of as a greater evil than the actual evil going on. There is a time to speak out, and a time to keep silence - Ecclesiastes 3:7.
An unattributed saying is, "Worry is interest paid on trouble before it's due." Not a biblical quote, but it rings bells with me regarding what Jesus said about not worrying as to what we will wear or eat or drink, for God knows what we need regarding material things, and will supply our needs if we "seek first his kingdom and his righteousness" (Matthew 6:31-33 N.I.V.) Jesus rebuked those who worried about such every-day concerns with, "O you of little faith" (vs. 30).
Nearer the point of your question is what he said to his disciples during a storm that his disciples said would drown them:
"You of little faith! Why are you so afraid?" Then he got up and rebuked the winds and the waves, and it was completely calm." (Matthew 8:23-27)
If ever there was a biblical example of not letting fear about a terrifying situation grip one, that is a good one. On both those occasions, Jesus highlights lack of faith as the underlying cause for fear and anxiety. Yet that's still not exactly what you're asking for. Try this biblical proverb:
"Do not fret because of evil men or be envious of the wicked, for the evil man has no future hope, and the lamp of the wicked will be snuffed out." (Proverbs 24:19-20 which is almost identical to Psalm 37:1)
But is anticipation of evil worse than the evil itself? you ask. It only would be if our anticipation caused us to succumb to the evil instead of standing firmly against it. And that would ultimately be due to our lack of faith in God dealing righteously with evil, and evil people. This is where the book of the Revelation builds up sufficient faith in God's sovereignty over evil things, and evil events, that would otherwise overwhelm God's people.
It takes faith in God to hold to such a fearless view of evil, trusting that "greater is he who is in you than he who is in the world" (1 John 4:4). John adds that fear has to do with judgment (fear of being judged) and not having been made perfect in God's love, for "There is no fear in love" (4:18).
However, a powerful biblical principle that applies to the evil of anticipating evil is that "everything that does not come from faith is sin" (Romans 14:23). As has already been pointed out, lack of faith will contribute to being fearful, therefore anticipating something evil whilst claiming that God is greater than all evil, would be sin if such anticipation of a real, looming evil caused us to doubt God and to be overcome by that evil. But being made perfect in God's love should help us to see looming evil with faith. And the book of the Revelation is full of looming evil against God's people, in all the centuries since Christ's ascension. Yet the prophetic account has been given to God's people to maintain faith in the sovereignty of God against all evil, and in his perfect love for those who have passed over from judgment to life, even before they die physically.
If we broaden "evil" beyond the moral connotation to things like hunger and other material needs:
an NT verse example is Matt 6:31-33, part of Jesus's teaching to trust God the Father because He surely love his human children more than the birds, the lilies in the field and the wildflowers:
31 “Therefore do not be anxious, saying, ‘What shall we eat?’ or ‘What shall we drink?’ or ‘What shall we wear?’ 32 For the Gentiles seek after all these things, and your heavenly Father knows that you need them all. 33 But seek first the kingdom of God and his righteousness, and all these things will be added to you.
another NT verse is 1 Pet 3:13-14, out of the theme of how we are blessed when we endure suffering for Christ's sake:
13 Who is going to harm you if you are eager to do good? 14 But even if you should suffer for what is right, you are blessed. “Do not fear their threats; do not be frightened.”
an OT verse is Jer 16:14-15, out of the fear that Jerusalem citizens had upon being conquered by the army of Babylon, where Jeremiah encouraged them that God will bring them back from what will be a temporary exile just as God had rescued them from Egypt several hundred years before. This can be applied by Christians to any suffering today: that as long as our hearts are in the right place, the evil we experience will pale in comparison to our future eternal life.
14 “But the time is coming,” says the Lord, “when people who are taking an oath will no longer say, ‘As surely as the Lord lives, who rescued the people of Israel from the land of Egypt.’ 15 Instead, they will say, ‘As surely as the Lord lives, who brought the people of Israel back to their own land from the land of the north and from all the countries to which he had exiled them.’ For I will bring them back to this land that I gave their ancestors.
All 3 verses above convey the message that although the anticipated evil can seem to be bigger than what we can handle, it is actually less than the actual evil if we trust that God is with us.
I have yet to find a verse that frames the above as a general principle. I think it will most likely be found in the wisdom books.