I don't believe that it was death itself that caused Jesus to say those words about the cup. Rather, all that was involved in His particular death - Him bearing the sins of His people, and thus being separated from His heavenly Father. He knew that death itself had no power over Him:
“15 “As the Father knows Me, even so I know the Father; and I lay down My life for the sheep. 16 “And other sheep I have which are not of this fold; them also I must bring, and they will hear My voice; and there will be one flock [and] one shepherd. 17 “Therefore My Father loves Me, because I lay down My life that I may take it again. 18 “No one takes it from Me, but I lay it down of Myself. I have power to lay it down, and I have power to take it again. This command I have received from My Father.”” (Joh 10:15-18 NKJV)