Last turn in Ticket to Ride: clarification
He can only claim a route with length 1 because he only has one train.
You will not magically get more trains because it is the last turn so all normal rules still apply.
By the rules:

Game End
When one player’s stock of colored plastic trains gets down to only 0,1 or 2 trains left at the end of his turn, each player, including that player, gets
one final turn. The game then ends and players calculate their final scores.

There is no mention of different rules for the last turns so normal rule apply.