He was only 27 in 1963, but he (teach) already at Galton University for five years.

He was only 27 in 1963, but he had already been teaching at Galton University for five years.

for five years - указывает на продолжительность действия (Continuous)

already - указывает на перфектное время (Perfect)

was - прошедшее время, для согласования и 'teach' должно стоять в прошедшем (Past)

Значит, нужное время - Past Perfect Continuous.