Can I use Arduino to build a single seated quadcopter?

Although it adds a significant layer of complexity, you may consider developing the Arduino / cockpit controller as one project - without 'actual' motors etc.

Then in place of the power drivers & motors, develop a desktop program to take the flight control driver signals - and display a visual simulation of the resulting 3D flight/motion on the PC screen.

This saves you making & wrecking the only $$ prototype (you will!), and reinforces your understanding of the physics, algorithms & flight dynamics required to perform all the functions you will need.
Your teacher will be very impressed!

Once this is built - you can tweak any variables - in real time without risk - and ultimately build the actual craft to hold the power, drivers and motors (and you!)