Yes, Paul did know that Timothy was going to live through the "last days." And, Yes, the "last days" did refer to the "end times." The problem is that most people do not understand what the "end times" and "last days" mean in the scriptures. When we critically examine the scriptures without the preconceived, taught biases of church dogma they become clear.
Beginning with Gen. 49:1, where Israel was blessing his children before he died, He said:
" And Jacob called unto his sons, and said, Gather yourselves together, that I may tell you that which shall befall you in the last days." (KJV)
We first notice that the blessings concerned the sons of Israel, the tribes of Israel, and then notice that the last days were of the end of the tribes of Israel. No where in this discussion is any warning of an end of the world scenario.
Gen. 49:9-12, concerning the "last days" of Judah:
"9 Judah is a lion's whelp: from the prey, my son, thou art gone up: he stooped down, he couched as a lion, and as an old lion; who shall rouse him up?
10 The sceptre shall not depart from Judah, nor a lawgiver from between his feet, until Shiloh come; and unto him shall the gathering of the people be. (KJV)
The sceptre was to pass from Judah to Christ (Shiloh) during the last days of Judah's existence. The scepter passed from Judah to Christ at His death on the cross in 30-31 AD.
In Gen. 49:24, discussing the end of the tribe of Joseph:
" But his bow abode in strength, and the arms of his hands were made strong by the hands of the mighty God of Jacob; (from thence is the shepherd, the stone of Israel:)" (KJV)
The shepherd and the stone of Israel was the appearance of Christ. The last days were the last days of the tribes of Israel when Christ would be manifest on earth!
We can now see some very clear time statements which John the Immersor (Baptist) and Christ both said in the New Testament.
"And saying, Repent ye: for the kingdom of heaven is at hand." (KJV)
Being "at hand" meant the time was near, it was very close, and would be there soon after John's prophesy when he spoke those words in the first century AD.
Matt. 3:10, John told the Pharisees and Sadducees who came to watch,
"And now also the axe is laid unto the root of the trees: therefore every tree which bringeth not forth good fruit is hewn down, and cast into the fire."
The axe being laid to the root of the trees indicated that the time was near that their tree (nation)would be cut down. Trees represented nations, or strong men in the OT (Isa. 10:15-19; 37:24; 61:3; etc.) John was telling them that they were about to be destroyed.
" For all the prophets and the law prophesied until John." (KJV)
Christ told them that all of the prophets and all of the law were UNTIL John. John's coming to clear the path for Christ, the Messiah, was the time of which all the law and the prophets spoke. And, when did John come to them, but during the last days of the tribes of Israel - in the first century A.D.
In Matt. c.10, Christ was telling His disciples how they should go, what they should do and say when He sent them to the cities of the "lost sheep of the house of Israel" (vs. 6), and He told them in vs. 23:
"for verily I say unto you, Ye shall not have gone over the cities of Israel, till the Son of man be come." (KJV)
As He was standing before them when He spoke those words, Christ spoke of His second appearance (coming) in that generation of the first century A.D. They were not going to finish preaching the gospel to the cities of the tribes of Israel before He came back to them.
In Matt. 16:27-28, when speaking to His disciples Christ said:
"27 For the Son of man shall come in the glory of his Father with his angels; and then he shall reward every man according to his works.
28 Verily I say unto you, There be some standing here, which shall not taste of death, till they see the Son of man coming in his kingdom." (KJV)
Jesus told the disciples that some of them would live to see His coming in His kingdom. That very clearly placed His second appearance to that generation of the first century A.D.
Matt. 23:35-36, prophesying judgment upon the scribes and Pharisees in Jerusalem,
" That upon you may come all the righteous blood shed upon the earth, from the blood of righteous Abel unto the blood of Zacharias son of Barachias, whom ye slew between the temple and the altar.
36 Verily I say unto you, All these things shall come upon this generation." (KJV)
After His resurrection while speaking to His disciples the third time, Christ told Peter that John would remain to see His coming:
"Jesus saith unto him, If I will that he tarry till I come, what is that to thee? follow thou me." John 21:22 (KJV)
Gal. 4:4 affirmed the time:
" But when the fulness of the time was come, God sent forth his Son, made of a woman, made under the law," (KJV)
Heb 1:2 affirmed the time:
"Hath in these last days spoken unto us by his Son, whom he hath appointed heir of all things, by whom also he made the worlds;" (KJV)
Christ spoke to the people in the first century A.D. during His earthly ministry, and the scripture said that He spoke to them in the "last days", therefore the time of the "last days" was the first century A.D.
1 Pet. 1:20 affirmed the time again:
"Who verily was foreordained before the foundation of the world, but was manifest in these last times for you," (KJV)
Christ was manifested during the fulness of times, and He was manifested in the "last times", and that was in the first century A.D.
As the physical world is still here, then the "last times" and the "end times" of the scriptures were not speaking of the "end of the world". The scriptures were always speaking of the end of the old covenant, the end of the old Mosaic animal sacrificial system (end of sins, Dan. 9:24), and the end of the tribes of Israel.
His second appearance promised to that generation (Heb. 9:28) was His coming in judgment against those who had crucified Him - the Jews - and He told them He was bringing His kingdom with Him (Matt. c. 23, 24, &25). The destruction of Jerusalem, and the destruction of that earthly temple with all its animal sacrificial system was the fulfillment of His second coming and it happened in A.D. 70.
The Bible prophesies never spoke of the end of the world. In fact, it implies the opposite. "...world without end. Amen." (Isa. 45:17; Eph. 3:21.)
"...and the Lord said in his heart, I will not again curse the ground any more for man's sake; for the imagination of man's heart is evil from his youth; neither will I again smite any more every thing living, as I have done." (KJV)
We have God's assurance that He will not bring a world-wide judgment upon the earth as He once did. The end times of the NT prophesies concerned a local judgment, not a world-wide judgment, and it was aimed at the those who crucified their promised Messiah, and persecuted His church. It concerned the end of the tribes of Israel, just as Jacob had prophesied in Gen. c.49.
See the posts at my blog ShreddingTheVeil Parts I - X of It's Not The End of The World, and Did Christ Lie to His Disciples, as well as The Whore of Babylon, The Beast of Revelation, The Seventy Weeks of Daniel chap. 9, and others.
Further readings can be found at: PreteristCentral, and specifically on the "latter days" here.