Hi Bjorn
If the buttons are acting as ON and OFF control for a contactor type starter then you will need to check the coil voltage, it may be 380v in which case you will need to replace with a 220v type.
You will also have to modify the control wiring to include a Neutral conductor to replace the coil return to the original phase connection.
The incoming 220v supply to the starter will be connected in a different form to the 3 phase 380v supply so it is best to obtain a schematic (circuit diagram) for the type of starter fitted.
Further to the above the overload (if fitted) will most likely need changing to a higher rating to suit the new motor FLC.
With the cost of a coil and overload unit it may be wise to consider a new starter to suit.
If you post some pictures of all the parts concerned it may be possible to help with positive information rather than guessing.
Emgee