Triggers not reflecting on procedure
Your trigger is designed to only fire after an insert not an update. Try changing the code to:-

create trigger flightinput2 on reservation AFTER INSERT, UPDATE as 
begin
   update reservation set Departure_Airport='LDN',Arrival_Airport='KUL' where Flight_Code='EM0088'
end


EDIT:- As pointed out, the code above will update all records within the database that have a flight code of EM0088

To prevent this you could update the code to:-

 create trigger flightinput2 on reservation 
 after insert, update
 as 
 begin
      update reservation 
        set Departure_Airport='LDN',
            Arrival_Airport='KUL' 
        from inserted ins
  inner join reservation r on ins.PrimaryKeyColumn = r.PrimaryKeyColumn
       where r.Flight_Code='EM0088'
  end


Just change PrimaryKeyColumn to the PK of your table.

EDIT2 - If you want to change values in the table based on flight codes you could do this:-

  create trigger flightinput1 on reservation 
  after update
  as 
  begin

  declare @code char(10);

  select @code = 
    ins.flight_code
  from inserted ins

  if @code = 'MH0007'
    update reservation 
    set Departure_Airport='XXXX',
    Arrival_Airport='KUL' 
    from reservation r
    inner join inserted ins1 on ins1.resid = r.resid 

  else if @code = 'EM0088'
    update reservation 
    set Departure_Airport='LDNXX',
    Arrival_Airport='KUL' 
    from reservation r
    inner join inserted ins1 on ins1.resid = r.resid 

  end