Wow, that lock is really unclear about how it works. Is the solenoid already in it? If so, what is the voltage it requires? If you are sourcing your own solenoid, there are lots to choose from at different voltages. There is no reason you really HAVE to need 18v (the two 9V batteries) as there are solenoids that operate at a much lower voltage. So in that case, you could just power everything from the same voltage. Though, one suggestion is that solenoids usually need a lot of current quickly. Either a battery, or a very large cap to store some juice should be used for this, or you may find your arduino resetting when the solenoid activates.
You say "I have a keypad that unlocks a solenoid..." does that mean you already have a keypad that does this, or is that your project? I ask because they do sell keypads with all that built in (I have a couple.)