"I was about to board the plane."
Translation:J'étais sur le point de monter dans l'avion.
You can "monter l'avion" but it means that you are putting a model airplane together, or even a real airplane.
You can also "monter quelque chose to an upper position", like "monter des valises dans la chambre" (carry suitcases up to the room).
To board is "monter à bord de l'avion" or "monter dans l'avion".
With the usual only-an-advanced-learner-not-a-native disclaimer, that sounds to me like your climbing on top of the plane, rather than getting into it.