Anthony Mann, starring Tom Conway and Ann Rutherford.
Conway, who sounds a great deal like his real-life younger brother George Sanders, plays an amnesiac who staggers in front of a taxicab driven by Patty Mitchell (Rutherford). Sympathetic Patty is intrigued by the mystery surrounding the man she had nearly run down, and together they spend the evening unraveling the man's lost identity and his possible connection to a murder.
The lovely Rutherford is very appealing as Patty, and her energetic performance has a great deal to do with the movie's success. Conway is somewhat bland but pleasant, and he has a nice camaraderie with Rutherford. The film's cast, the briskly told story (which clocks in at just 68 minutes), and very attractive black and white photography by Jack MacKenzie all combine to make this an entertaining little movie.
Prominent in the supporting cast is one Bettejane Greer, who shortly thereafter would shorten her name to Jane Greer and go on to film noir immortality in OUT OF THE PAST (1947).
The cast also includes Jean Brooks, Richard Lane, Lester Matthews, Roland Drew, and Emory Parnell.
IMDb and Leonard Maltin both offer the info that this film is a remake of TWO IN THE DARK (1936), which starred Walter Abel, Margot Grahame, and Gail Patrick. (Update: Here's a review of TWO IN THE DARK!)
TWO O'CLOCK COURAGE is not available on DVD or VHS, but it can be seen on Turner Classic Movies, which has the trailer available here.
January 2014 Update: I had a wonderful opportunity to revisit this film on a big screen at UCLA! I enjoyed the movie even more, having become much more familiar with actors such as Jean Brooks and Richard Lane in the time since I first saw it.