I thought for a P-FET the current passes through the drain and source until a positive voltage is applied to the gate which breaks the connection between the drain and source?
No that is not how FETs work.
So technically, when no voltage is applied to the gate, it will act like an closed circuit and when the arduino's digital pin activates the gate, the circuit will be broken?
No. Enhancement mode FETs work by lowering the resistance of drain to source due to the electric field applied.
You could use a depletion mode FET but I think these are rare nowadays, and anyway it will not do what you want.