When does the FMS in the 737NG switch to approach mode?
If the 737 is like most other FMS installations you will find the approach will activate at the capture fix.  It is typically the waypoint just before the FAF.  If you were flying into KMEM RNAV RWY 36C the capture fix would be NESBT and the FAF would be GINIE.
In the FMS that I use, the capture fix is typically loaded twice into the flightplan.  Once from the enroute structure and once again for the approach after the approach name.

NESBT
** RNAV 36C **
NESBT
GINIE