I am not sure what the requirements for flight by reference to instruments only were in 1959, however an altimeter and compass are not the minimum to keep the Earth from rising and smiting thee...
Yes, the compass is required and yes, you need a sensitive (able to read 20 foot increments as defined by the FAA) altimeter, one requires at a minimum a gyroscopic turn coordinator and as a luxury item a gyro compass. A gyroscopic artificial horizon (pitch and roll gauge you referred to?) was not originally a requirement but I'm not sure about 1959.
I do know from the "partial panel" practice that I did during my initial training (before multi-functional displays and Garmin GPS Whatever Number they are up to now) that after losing an artificial horizon it is possible to keep an airplane right side up with a whisky compass, altimeter and turn coordinator but you absolutely need that one gyroscopic instrument (turn coordinator) to survive.
In the instrument qualified aircraft of the day while other gyroscopic indicators were usually powered by vacuum sources, typically the last-ditch, absolutely must have turn coordinator was powered electrically. Lose the electrical, well, you still should have and artificial horizon and gyro compass. Lose the electrical system and you lose the turn coordinator.
Redundancy is key to life and it is what ultimately brought (as bad as it was) the crew of Apollo 13 back home to a safe splashdown...
Thus, even an aircraft that you described earlier as a death trap had redundancy established. Were the expectations the same in 1959 as they are now? Oh hell no! Those poor people couldn't have even imagined a flip phone let alone the latest Samsung or I-phone!
Yes, the pilot was not qualified to handle the flight in question regardless of how the aircraft was equiped. It was a sad but predictable outcome...
As far as I know the only gyro on the Spirit of St. Louis was the turn coordinator. For a primary compass the Spirit was equipped with an Earth Inductor compass and I believe no gyro compass was installed...
scroll through the pictures of the Spirit of St. Louis in Washington, DC-
airandspace.si.edu
Charles Lindbergh was a frelling genius...