IIRC his "scarred and deformed" appearance after his fight with Mace Windu was said to be his true appearance that the force lightning uncovered. Given that such deformation usually results from decades of calling on the powers of the dark side and given the extent of his deformation he must have been a dark side user for quite some time.
His master, Darth Plagueis died 50 years before Episode IV and its reasonable to assume that he had spent at least ten years training Palpatine before that and that Palpatine was at least 10 when said training began, meaning that Palpatine was at least 50 when he became Emperor and at least 70 when he died in ROTJ.
Wookiepedia gives his birth as 82 years before Episode IV making him 86 when he died.