Palpatine became ugly all of a sudden due to the effect of his own "Force Lightning". It has nothing to do with old age or faster aging. In the movie Star Wars Episode III: Revenge of the Sith, when Palpatine attacked Mace Windu with Force Lightning, Mace Windu used Vaapad to reflect Force Lightning to its source.
From the Revenge of the Sith novelization by Matthew Stover:
Lighting blasted the clouds above, and lightning blasted from
Palpatine's hands, and Mace didn't have time to comprehend what
Palpatine was talking about; he had time only to slip back into Vaapad
and angle his blade to catch the forking arcs of pure, dazzling hatred
that clawed toward him.
Because Vaapad is more than a fighting style. It is a state of mind: a
channel for darkness. Power passed into him and out again without
touching him.
And the circuit completed itself: the lightning reflected back to its
source.