Jesus' response was very human, to fear death and struggle. However, He doesn't stay in fear. His relationship with the Father allows Him to take the next step of faith - to humbly set Himself aside for the purpose which He must perform.
Instead of staying in the perspective of the world, "Father take this cup from me," He continues to a Godly perspective in one breath, "not my will, but Yours be done."
No one wants to face the unknown, especially death; however, the next step of faith is to believe in God's purpose. That is to know that something great is coming out of the trial that will make the struggle worth the sacrifice.
This does not have to be applied to death alone, but even struggles of finance, sickness and even emotional battles. You can ask God to take the pain away, but then appreciate God's Almighty glory and take courage in "not my will, but Yours be done." That faith with make not only this moment of trial more endurable, but also in the future will be a permanent reward for all who are faithful to God the Father.