Can I use a mosfet as a switch to power arduino?
Yes, that will work, but you don't need the 50Ω resistor as it will only waste power.

The second thing is you may not need to use this circuit at all! Why? because you can get an arudino to use very little current in a sleep mode, like 10uA of current. The supply current is at least 3mA for an NE555 and the 555 timer needs to be operating all of the time. The aruidno only needs 10uA to sleep so a 555 timer wastes 300x more current.

And you can simulate circuits like this:
 (you can click on edit below the image and then click simulate) 
I have simulated the load as 10Ω, which would be a peak load of 500mA for an arudino.







simulate this circuit &ndash; Schematic created using CircuitLab