The only thing you need for this battle is Donatello, so keep him alive for the battle.

1. Enter the boss room as Donatello (As mentioned in the prerequisite)

2. Before Shredder fully spawns jump to the right platform.

3. As soon as Shredder spawns hit him twice to knock him back to the floor.

4. Shredder is now in an infinite loop of trying to jump out. Now it is time to beat up Shredder by staying on the top platform and hitting him with Donatello's staff. If timed right you should always hit Shredder.
Congratulations! You have not only beat Shredder but also beat TMNT on NES too.