I need this motor in a device I am manufacturing. I want to buy/build the "Customer Circuit" below. I would like to set the speed manually or using an adruino uno. How should I proceed?

 

Welcome to AAC!
What is the motor type/specification?

 

Thank you!

This is an AC motor for a centrifugal fan.

 

I'm not an expert on AC motors, but am aware that some types don't lend themselves to easy speed control , or have only a limited range of control.
Phase control of universal motors is relatively easy, but I doubt your fan motor is that type. Does it have a start/run capacitor?

 

Small fan motor of the shaded pole or PSC (Permanent Start Cap) motors under 0.5hp can usually be controlled by Triac control, e.g. ceiling fans etc.
The 480w motor shown would seem to fall into that category.
There is a few off the shelf that would do this and I am sure there is an Arduino solution out there by now.
I don't use Arduino so cannot help there.
Max.