Is it feasible to use a servo to rotate and counter rotate the door?
If the servo has the torque needed to turn the knob.
If so what would I need to construct it?
A bracket on which you can mount the servo and some way to connect the servo bellcrank to the knob. Possibly a dome to protect the servo from rain.
An arduino unit - which one would be suitable?
Any. The UNO is the most common.
A shield?
No need. The servo has a built-in motor controller.
Something to power the servo
A 7V 1A power supply should serve both the servo (7V) and the Arduino (via Vin).
Note: For time-based operations you should probably get a Real-Time Clock (RTC) chip. The Arduino UNO is not great for long-term timekeeping.