What is the proper way to oil up a Model 137 air pistol?

First of all DO NOT use regular oil on an airgun, Always use "air gun oil" like RWS or Crosman pellgun oil. Regular oil will eat the seals. You did not state who manufactured this pistol. I'm going to guess 'Crosman."

Oil the hinges, Put a drop of oil in the breech and the barrel to coat the inside. Put a few drops on a small rag and wipe the pistol down. That's all that is needed.