I moved your post from the Radios forum to this one, which deals with acoustic/wind-up phonographs and gramophones.
First of all, that's
not a Sonora phonograph. Somewhere along the line, someone stuck a Sonora brand sound box (reproducer) from the early to mid-1920s onto the end of the tone arm, probably to replace the original one that was damaged or lost.
The original sound box would be more like this one:

What you have is known as a "mechanical field phonograph." (Do a Google search for that term.) These were made by the Pacific Sound Equipment Company, Hollywood, CA. and were distributed to all branches of the service, along with special "V Discs" that contained popular songs.
Was this one was ever used on-board the USS Arizona? Probably not. Most of these would have been produced
after the US entered the war, concurrent with the issuance of the V Discs --
http://en.wikipedia.org/wiki/V-Disc . There may be a slip with the manufacturer's date and serial affixed to the motor-board, under the turntable, or inside the cabinet near the motor.
Along this line, the phonograph would have been the property of the ship and not one of the crewmembers, so it probably would not have made its way off the USS Arizona--unless, of course, it was stolen.
As for "value," I've seen these sell for as little as $75 (on craigslist) and as much as $400 (on eBay)--depending on condition. Many of these were sold by military surplus stores after the war, and these often turn up in like-new condition today. The one you have is in what I'd call average condition--the case has a bit of corrosion and some paint loss, and it's missing its original sound box.
HTH,
JDS