Should he offer his seat to an elderly person and risk accidentally touching a woman if he stands up on the bus?

Praise be to Allah.

If the situation is as described, and standing up means that you will be crowding with women or they may touch you, then stay in your seat so as to protect yourself from the causes of fitnah and falling into evil. 

And Allaah is the Source of strength. May Allaah send blessings and peace upon our Prophet Muhammad and his family and companions. End quote. 

Standing Committee for Academic Research and Issuing Fatwas