For lo, the wicked bend their bow, and make ready their arrows upon the string, that they may secretly shoot at them, which are upright in heart.
Psalms 11:2 GNV
Reading several of these very first chapters of the Psalms we know there were downright enemy nations that opposed Israel and came against them with arms such as bows and arrows, spears, darts (big deadly ones that vould pierce the heart), even arms described as "engines" which may have been catapults, trebuchets, or even something else able to otherwise breech the walls of a castle or fortress. But a second look at this chapter, even in this verse it is very apparent lthat the one bending the bow is who is considered to be the enemy. It is the one against the upright in heart. It is worth knowing and understanding that who our enemy is isn't defined by ourselves as much as it is by those who oppose us.
Throughout the days of the ministry Jesus had did He oppose anyone? Most certainly. He challenged every false view of God and His Word, that Word being Jesus Christ Himself. He was upright in heart and several times Jesus pointed out the true intent of the heart of others when He said that the Jewish leaders were going about to kill Him (John 7:19). Later we read that they were actively seeking a means to have Jesus killed. Even spoke of killing Lazarus as it became so substantiated that he was definitely dead and Jesus had called him from the grave, and then because of that he was one of the greatest proofs that Jesus was the Son of Man, and the Son of Man having power to forgive sins (Matthew 9; Mark 2; Luke 5). Jesus enemies defined themselves and even Judas Iscariot, one of His disciples got caught in their trap with the money they offered. And in Judas is seen what Jesus taught in the parable of the tares (Matthew 13:24 - 30, 36-43). Of Judas we are told that Satan, the devil entered into Judas Iscariot. In the explanation of the parable of the tares Jesus said specifically that the one, the enemy who sewed the tares was the devil. The ones offering the money to betray Jesus were simply minions of his choice, but Judas was the very piece the devil sought to be the enemy.
So thinking of Psalm 11 as a prayer Jesus prayed, as dark as it is, yes, it is reasonable to understand that Gethsemane was not the only time during His ministry the dark burden of opposition in ministry, especially those who directly opposed Jesus was very overwhelming. When you think of a Holy God living as the Word who became flesh dwelling among men it might seem more realistic that such burdens could have become a daily prayer for Jesus because of the increasing mounting opposition by His enemies.
When applying to our own lives in learning to pray possibly these psalms Jesus began His sermon to His disciples with what is called the Beatitudes being capped with "Blessed are you when men shall revile you and do all manner of evil against you for my name's sake". (Matthew 5) Much later Peter would write:
For what praise is it, if when ye be buffeted for your faults, ye take it patiently? but and if when ye do well, ye suffer wrong and take it patiently, this is acceptable to God.
1 Peter 2:20 GNV
Jesus knew when He was tempted by those opposed to Him He hadn't done wrong. But Peter knew all to well in so many things when he was buffeted for his own faults including having denied Jesus three times. But going forward after the resurrection and ascension in Acts 3 and 4 John and Peter where going into the temple, healed a man in the name of Jesus, ended up in prison, were beatened, and threatened not to preach in the name of Jesus; and yes, that was by the same group who took up arms against Jesus and He let them put Himself to death. What did John and Peter do? Went their way praising and rejoicing that they were counted worthy to suffer for His name. Their prayer even was foremost and mostly in acknowledging who God is and His greatness. The only thing they even asked for themselves was strengthening by the Holy Spirit to go out and continue to speak in His name. While others armed with threats told them not to speak in this Name, the disciples arms, their strength lay only in the Holy Spirit. Not only would Jesus use a psalm like this in seeking His Father, but later so would His disciples.