Electric strike with Arduino circuitry?
To be clear, what you posted is a wiring diagram, not a schematic.  A schematic of your circuit would look something like this:




  Would a 2N3904 NPN transistor and a 1kΩ resistor work together in this
  scenario


1kΩ is probably fine.  We don't know the peak and holding current for your magnetic strike (they rarely publish such information from my experience).  The 2N3904 is rated for 200mA, likely not enough for your strike.  I'd guess that the strike will take around 1A at 12V, so you may want to use a relay to handle the current.  For that, you can use your 2N3904, just use it to drive the relay coil (5V, powered from your Arduino, or 12V, powered from your external supply), and use the relay to control the strike.   This discussion may be helpful.


  What would happen on the Arduino if you had more than two wires that
  needed to be grounded


It's not quite clear what you're asking.  The Arduino doesn't care how many ground wires are attached (they'll all be at the same electrical potential, or 0V).  What the Arduino may care about is how much current needs to flow into its ground pins.  As long as you're not passing the current from the strike through the arduino board, you should be fine.




  Is the diagram that I have suitable for the strike, or am I missing
  anything.


Note the flyback diode in the schematic.  Even if you go with the approach of using a transistor to drive a relay and using the relay contacts to power the strike, you'll want a flyback diode across both the relay coil and across the strike itself.  

The strike is basically a big inductor - once current starts flowing, it wants to keep flowing.  Most strikes have a MOV snubber integral with them to suppress spikes, but a reverse-biased diode will do a better job of eliminating transients.  A lot of strikes are made to work on either AC or DC; a MOV works for both, but the reverse-biased diode can only be used when applying DC.


  The power supply I linked to.


Should be fine, based on the limited information we have.  If the strike needs 2A or less, it should work.


  Can you tell if it has two wires or one?


Assuming that you're referring to the strike, I'd guess that it will have two wires, but there's only a couple ways to tell for sure:  Ask the reseller or order one.


  why do strikes such as this one have 4 wires, instead of 2


So they can be used in either 12V or 24V applications.  For 12V, connect the two coils in parallel; for 24V they're connected in series.  There should be a (sometimes very small) datasheet that comes with the strike that has connection instructions.

I'd recommend something like the following: