Is bodybuilding permissible to make one's appearance more masculine and to maintain health, and how can one avoid arrogance in it?

There is no harm in practicing bodybuilding if the practitioner avoids the prohibitions usually associated with it, such as exposing the 'awrah (private parts), and if it does not lead to neglecting an obligation like prayers.

Arrogance (kibr) is defined as rejecting the truth and disdaining people. That is, pushing away and denying the truth out of pride and tyranny, and despising and looking down upon people.

To avoid arrogance, do not disdain people, do not reject the truth, and do not consider yourself better than others because of what Allah has bestowed upon you. Rather, attribute all of that to Allah's grace and use it in what pleases Him.