When purchasing the 13th technology, can you use its power right after doing so?
No you can not.

From the FAQ file on BGG here the following Q and A is included.


  Q: When do Technology benefits trigger? A: Each Technology triggers at
  most once during your turn. Most Technologies trigger after performing
  the Main action on the board (or boards) listed at the bottom of each
  Technology tile. No Technology will trigger during the turn when it
  was acquired.


I've tried to find evidence to back this up in the rules printed though and could not.