Electronic – arduino – Help with simple Arduino/Transistor/Solenoid circuit

In general, you want the flyback diode as physically close to the inductor it is catching the current from as possible. Long leads mean inductance, which slows how fast the current can build up in the diode, which makes large voltages at the solenoid.

If the only thing that is at the end of the long wire is the solenoid with the diode and driving circuitry at the other, then the diode will still protect the driving circuitry well enough. However, the possibly high voltage transients at the solenoid end of the wire can cause radio interference, and in extreme cases arcing.

With long wire to the solenoid, I'd want two diodes. One at the solenoid to limit voltages on the wire, and another close to the driving circuit to protect it. The inductance in the long wire can cause kickback too.

No, this transistor cannot be expected to do this job for long, if it can do it at all.

From the datasheet, look at the "On characteristics" on page 2. First, its free air power rating is 0.625W, which means Vce had better be 1.25V or less at 500 ma.

Then, gain (hFE) is shown at different Vce voltages and currents. But significantly, not shown at Vce=1V and Ic=500mA, suggesting that the transistor is not rated to work under those conditions.

Finally, the CE saturation voltage is shown as 1.6V at 500mA, which exceeds the power rating shown above. You will be able to get away with that for a few seconds on a very low duty cycle. The MPS2222A would be a better choice, its Vce(sat) is shown as 1V here.

But...the above condition is achieved with Ib=50mA. This almost certainly exceeds the current available from your Arduino output pin.

If you are content to briefly overrate the device's power ratings, you could overcome the base current limitation using a second transistor as an emitter follower, to drive the base current you need.
(schematic editor isn't loading this morning, sorry)