Both destinations are served by Amritsar (ATQ). Since there's no flight route between them, the best way to travel is by road.
Golden Temple Amritsar
Fly to Amritsar (ATQ), then travel by road to Golden Temple Amritsar. Taxis, buses, and private cabs are available from the airport.
Amritsar
Fly to Amritsar Airport (ATQ), then travel by road to Amritsar. Local transport options include taxis, auto-rickshaws, and state buses.