Pain on the rift side behind breastbone. More like pressure than pain. None on left at all. Something to be worried about?

Doctor can evaluate. Any person with chest pain has to go to his doctor to be checked for a heart attack. That means not waiting at home to see if it becomes a bad heart attack. It may turn out to be stomach acid reflux, a lung problem like pneumothorax, or a bone or muscle problem like costochondritis, but the heart attack kills people. As the saying goes, "better safe than sorry", so a visit to the doctor is wisest.