Are the batteries rechargeable ?
I ask because if not then they will provide 6V when new which can be an awkward voltage for the Arduino. Too much to feed directly to its 5V pin and not enough to feed to its onboard voltage regulator but not the end of the world. One solution would be to use a buck converter to increase the voltage supplied to the Arduino to say 8V and feed its voltage regulator with that but other methods are available
A Uno sounds like overkill, at least in size, to do what you want. Smaller Arduinos such as a Nano would make the perf board solution much easier and neater
Of more concern is reading the button on the toy. It is likely that one side of it is connected to either the negative (GND) or positive battery supply. Thus, when closed, the other switch contact will be taken to GND or positive. Are you able to establish how the switch is wired ?