90% of our equipment is rated for 50Hz or 60Hz, but I have learned not to take a chance with the rest. My solution is to use a GoHz solid state AC power supply to convert the 120V 50Hz to 240V 60Hz, then use an inverter to convert the 220V 50Hz to 110V 60Hz. If your apartment is wired for 240V 50Hz, I am assuming you can find 240V 50Hz AC to 110V 60Hz converters (~10A) on GoHz.com, then order some 1kVA pure sine wave frequency converter which should be $500-$1000 range. You will have 2 boxes instead of 1 and it is not the most electrically efficient configuration, but is the only way I know of that works with a reasonable price.
Alternatively, taking a motor from 60Hz to 50Hz will reduce the capacitor effects. This will result in possibly lower running and starting torques. There are transformer based solid state frequency converters, however, the problem tends to be that they are rated to work with a given load range. In the case of a light load such as you will have in a radio alarm, the voltage often goes above the rated voltage so will end up damaging the device or causing it to over-heat. Often the best way to convert low power devices is to replac the 50Hz to 60Hz frequency converter. If you are lucky it has an external power adapter, in which case replac that with one that has the same output voltage, but with a 220v 50Hz input voltage, more on GoHz.com