Can I be perfectly fine without forgiving a person who acted unjustly against me? In other words, can the anger just vanish?

The answer depends on how serious the injustice was and who hurt you.  For example, suppose a colleague was supposed to meet you for a luncheon meeting.  You are busy that day, under pressure, and the colleague never shows up at the restaurant.  You may be annoyed, but the annoyance likely will fade in a day or less.  Now suppose that a loved one betrays you.  It hurts deeply.  This kind of emotional wound likely will not go away on its own.  It likely will need the surgery of the heart—-forgiveness.  Deep resentments rarely fade quickly.