How to trigger 50v solenoid with 120v AC
The clock you show is a simplex synchronization clock.
The clutch is driven by pulsating DC and activated when the clock is in sync mode (turns fast to a fixed endpoint).
The 'valve' shown in the schematic is a Thyratron ....this is very like an SCR, but has a high on voltage typically in the 60-90V range. 
When the input sync signal is present (typically 3-4kHz) then the Thyratron is triggered, pulls in the clutch and the clock speeds up till it reaches the sync time. 

To replace the circuit you could use an MCU to receive the sync signal and trigger something like a MOC3020 on only the positive part of the AC cycle. Since you would have a full half cycle aplied to the solenoid you should put a series resistor about the same value as the clutch resistance in series.