Captain Clarence Oveur only makes it about halfway through the movie before falling violently ill at the hands of the infamous fish dinner. But Oveur still gives us some memorable moments before it's Oveur—er, over.
Peter Graves, who plays Captain Oveur, was famous at the time as a dramatic actor with an impressive list of serious roles to his name, including Jim Phelps in the "Mission Impossible" TV series. In fact, due to this reputation, Graves was reluctant to take the part. (Source)
In typical Airplane! fashion, Graves plays this character just like he would any other: completely straight. Like everyone else in the movie, Captain Oveur goes about his business with no awareness of the abundant ridiculousness, a reality that's heightened by Graves's track record and experience playing serious characters with supreme gravitas. Because of this, Oveur pulls off lines that nobody else could.
And with a crew named Roger, Unger, and Victor, he's got the best puns in the film.