I think 1 Thessalonians 4 has one of the more obvious references to a rapture-like occurrence:
1 Thessalonians 4:13-18 wrote:13 We do not want you to be uninformed, brothers, concerning those who are asleep, so that you will not grieve like the rest, who have no hope. 14 Since we believe that Jesus died and rose again, in the same way God will bring with Him those who have fallen asleep through Jesus. 15 For we say this to you by a revelation from the Lord: We who are still alive at the Lord’s coming will certainly have no advantage over those who have fallen asleep. 16 For the Lord Himself will descend from heaven with a shout, with the archangel’s voice, and with the trumpet of God, and the dead in Christ will rise first. 17 Then we who are still alive will be caught up together with them in the clouds to meet the Lord in the air and so we will always be with the Lord. 18 Therefore encourage one another with these words.
Rapture can be considered to mean being "caught up" or "carried away" in addition to its official definition of a feeling of "intense pleasure or joy."
Oxford Dictionary wrote:Origin
late 16th century (in the sense 'seizing and carrying off'): from obsolete French, or from medieval Latin raptura 'seizing', partly influenced by rapt.
That said, the passage could mean a number of things, and as Mech pointed out, the concept itself isn't very old at all.