Is the job of a manager to do the right things or to do things rightly?

They should consider doing both. It's always nice to know doing the right thing is right. Know what I mean? There are times where people would just goof off and doing whatever please them but you got to look outside of the box at times.

Very simply, a manager is accountable for the performance of others. Instead of working alone, managers spend most of their time planning, staffing, directing and organizing the work of others. They focus on resource allocation, goal setting and performance appraisal. Managers may be responsible for the employees in a department or division, they may have direct reports, or they may lead a project or a cross-functional team with no direct reporting relationships

its both