Can a person who has no symphathy nor emphathy for other people still perform a selfless act towards those same people?

Yes, if that will boost their reputation among others.

This is a tactic narcissists and psychopaths use to make people believe they are good people.

The most obvious example of this is the people who work in charity and have no empathy at the same time.

Yes. Serial killers often have a very helpful and pleasant personality. BTK is a good example.

They can perform “the act” but it would not be selfless, it would be expedient at best, and for nefarious purposes at the worst.