How about 10 x 1.2v NiMh AA cells? I'm assuming the door strike is only operated for a small fraction of the time, otherwise the capacity of AA cells will be too small. Or a small SLA battery? The only disadvantage I can see is that you will need to connect more solar panels in series to get 12v to charge them.
If you have a large current load on the 5v supply then you might want to use a switching regulator to get 5v from 12v efficiently, otherwise you can use the linear regulator already on the Arduino.
If you stick with the LM2577, then I suggest you connect the door strike permanently to the regulator output, and switch the input to the regulator circuit using a PNP power transistor. That way, the regulator always has a load of 210mA (making the design easier), and draws no current when the door strike is off.