In what portion a non-GPS approach, can a pilot legally fly with GPS as primary navigational resource?
An appropriately installed GPS may be used until crossing the final approach fix on a non-localizer approach, and until the segment requiring navigation via the localizer on a LOC, ILS, SDF, or LDA approach. It's generally approved and legal at any other time.

During the missed approach, a pilot may switch to GPS immediately, unless the missed approach requires navigating by using the localizer, in which case the ground-based navaid must continue to be used until the procedure no longer requires navigation by reference to the localizer.

The FAA covers this in AC 90-108:


  An otherwise 
  suitable RNAV system cannot be used for the following:
  a.   NOTAMed   Procedures. 
  Unless otherwise specified, navigation on procedures that are 
  identified as not authorized (“NA”) without 
  exception by a NOTAM. For example, an operator 
  may not use a RNAV system to navigate on a procedure affected by an expired or unsatisfactory 
  flight inspection, or a procedure that is based upon a 
  recently decommissioned NAVAID.
  b.   Substitution on a Final Approach Segment. 
  Substitution for the NAVAID (for 
  example, a VOR or NDB) providing lateral 
  guidance for the final approach segment.
  c. Lateral Navigation on LOC-Based Courses.
   Lateral navigation on LOC-based courses 
  (including LOC back-course guidance) 
  without reference to raw LOC data.