On Shabbat, is it permissible to enter an apartment building elevator and wait until someone else enters the elevator and simply ride up to the floor that this other person goes to for his own benefit?

If this other person is a Jew then it forbidden to ride up the elevater with him since your gaining from a chilul shabbat (a Shabbat desecration). If this person is a Non-jew, in certain cases it can be permissible. you should ask a rabbi about your specific case. Rabbi Yitzchak Grinblat