Convert low power Alarm Clock from 120V 60Hz to 230V 50Hz
If you plug it in to 230VAC it will be destroyed, without question. 
They do not convert the frequency. 
Yes, I believe it will run slower on 50Hz. The timing motor (probably a synchronous motor) may also overheat. 
No the 100W converter (assuming a transformer type, which is usual at that power level) is fine for converting the voltage. 
Try to find an adapter specifically designed for clocks. Even if the frequency was converted it would not likely be accurate enough for a clock by default. It will not likely be particularly cheap, but your vintage clock radio is priceless, right?