postgresql : **ERROR: path element at position 1 is not an integer:**
Your json value is an array, not a single value. So you need to pick the array element you want to change by including the index in the array you pass to jsonb_set()
UPDATE temp_table 
  SET owner_property = CASE 
                        WHEN owner_outgoing_edge IS NOT NULL 
                          THEN jsonb_set(owner_property, '{0, OutgoingVertexid}', '&quot;&quot;')
                        ELSE 
                          jsonb_set(owner_property, '{0, OutgoingVertexid}', to_jsonb(owner_incoming_edge)) 
                       END