What is the ruling on someone who swore: "By God, I will not let him bring her down [i.e., land a plane]" while in a state of stress, then she was brought down for him days later? Does he break his oath if he lets her down later?

If you did not intend to make an everlasting oath that you would never ride the bicycle, but rather swore because of the rain, this is called "the context of the oath" (bisat al-yameen); which is the reason that led you to swear. If you doubt your intention, you refer to the context of the oath, and nothing is binding upon you, nor is expiation required if you ride it on other days. You must also repent if you lied by saying "I don't know" while you actually knew the bicycle's location.