That short lifespan demonstrates a defective battery. I believe that a lithium-ion battery is the type in the solar watches. Unlike most people think, sunlight isn't just piped in the watch and
voila! it runs. The battery runs the watch and the sunlight recharges the battery.
I believe that Casio solar watches are designed to be out of the light for at most 6-8 months or so while continuing to operate, at the least, the time function. Kinetic watches work slightly different than automatics and, of course, a quartz watch relies on a battery for its power. Each kind of watch has its differences, pro's & con's, and advantages & limitations.
Invest in a set of precision screwdrivers (even inexpensive ones are fine), a watch case back opener, a spring bar removal tool (for removing watchbands), a small pair of non-magnetic tweezers, some extra batteries, a set of extra spring-loaded watchband pins, and an extra inexpensive, universal-fit watch band (cordura/nylon/whatever strap). If you don't live at your BOL, consider a kit of duplicates there as well.
If having a functioning watch after TEOTWAWKI is important, then plan & prep just as you would with anything else. What kind of watch you have/use is a personal preference. Like anything else, think about it ahead of time and prepare.
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Now...it's very unlikely that anything with chronographs is going to be needed after the stinky stuff hits the fan. Any type of extreme precision isn't going to be necessary - for example, a watch that is kept accurate by radio signals from the atomic clock. Even a moderate degree of precision might not be needed so a digital watch might not be necessary. A rugged, extremely water-resistant analogue watch might be the most that one needs. I strongly doubt that "synchronization" is really going to be a vital component of survival.
How that rugged, extremely water-resistant analogue watch is powered is a separate decision. The power reserve for most modern automatic movements is about 40 hours (+/- 2 hours) so it must be a watch that is worn often in order to keep it operating. Solar watches require both sunlight and a battery. Kinetic watches require movement as well. Finally, quartz movements require only a battery for operation. The quality of quartz movements vary widely from the Chinese knock-offs to the Swiss-made. Remember, by Swiss law, a watch movement is considered to be Swiss if it has been assembled in Switzerland; it has been inspected by the manufacturer in Switzerland;
AND the components of Swiss manufacture make up for
at least 50 percent of the value, without considering the cost for assembly. Just because a watch's movement is labeled "Swiss made" does NOT mean that its parts were manufactured in Switzerland or that a lil' ol' fourth-generation watchmaker is putting it together in the back of an old watch shop. Notice that the third clause of the legal requirement does not mean just the "components" of the movement itself, but may be (and always is) interpreted to extend to the entire watch (movement, case, crown, and band or bracelet).
Style and features are the final, and least important, factors to be considered if things have collapsed. A day and/or date display might be helpful or might be worthless, depending on your preference. Luminescence would certainly be a benefit for seeing the time after dark. Most "glow-in-the-dark" hands are recharged by light, not self-powered by, for example, tritium (such as pistol night sights). There are some luminous dials (they appear white in daylight) that actually glow while the hands and hour markers remain dark. I believe that luminescence would be/is a very important feature to have in a watch.
This would be a good example of a heavy-duty watch that would/could survive the abuse of TEOTWAWKI:
http://professionalwatches.com/2014/04/victorinoxs-most-rugged-watch-ever-the-inox.html. It's just been released and MSRP is $525 and that is the street price right now.
Here is the details of the Ronda 715 quartz movement:
http://www.ronda.ch/en/quartz-movements/quartz-essentials/ronda-normtech-700/caliber/715/?tx_hfronda_product%5BsourceTarget%5D=&cHash=2aa447e6e44eb4fd26c6090e6ca27cfd. As a point of interest, this Swiss-made movement is readily available online for $10-15.
That's just off the top of my head....