Depending on the type of motor, some motors follow the mains frequency to determine the RPM, others are independent of mains frequency, this depends largely on the armature type, if it is a cage type then the speed depends on the rotating field and hence the frequency of the incoming supply, but this should not really be a problem, it is not a clock as such that will drift in time keeping.
Most important issue being the voltage, if a motor is designed for 110V ac then it should never be subjected to as much as 220V it will burn out in a very short time, further more if it is not a mains synchronous or a cage motor then it may even run far too fast and destroy itself.
Most motors used in mains hand held drills for example are not dependent on mains frequency, they have a wound armature, and and a commutator with brushes, their speed is controlled by voltage alone.
A step down transformer rated to the same wattage or more should be used to transform 220V down to 110V. (if the motor rating is say 100watts at 110v then you need to make sure that the transformer needs to be rated to supply 110v at 100VA or just around 1 amp current, ideally a 200VA ( 110v at approx 2 amps) as an example, transformer is more safer but will be more expensive, motors also tend to draw a huge amount of current when starting so by having a twice the rated power, you are covering from this stalled motor condition, but have the transformer fused by an anti -surge fuse of appropriate value to prevent overloading the transformer under stalled condition.