Откройте скобки в Past Simple или в Past Perfect Suddenly he (to remember) that he (not to ring) her up in the morning/
Suddenly he remembered that he had not rung her up in the morning
по-моему, RING - неправильный глагол.
ring - rang - rung
поэтому иоя версия такая:
He remembered that he hadn't rung her up in the morning.