IMO walking is the best way. If it's raining, you can walk one level down (use elevators/lifts and/or escalators at either end so you're not going up and down stairs), arrivals rather than departures, and be under cover as you're walking through a sort of parking garage. It's a very short walk, almost to the building next door although there's a gap were the roadway turns the corner.

An alternative is a airport free shuttle bus, but it tends to go around the wrong way for you, from T4 to T5, T6,....and around the loop to T1, T2, and T3 before TBIT. Walking will be faster and less hassle. Make sure you don't take a bus going to airport hotels or remote parking.