Does the Citizen AT8260-50E need a battery change?

No. Eco-Drive charges from any light source, so there is no battery to replace. Keep it in light regularly so it stays charged.

How accurate is a radio-controlled Citizen?

When it receives the atomic clock signal it stays within about a second. Without a signal it runs as a normal quartz watch, accurate to roughly 15 seconds a month.

Yes. The 10 ATM rating covers swimming and showering. Do not press the pushers underwater or use it for diving.

Why is my watch not syncing?

Reception is weakest indoors and at night in some regions. Place it near a window overnight, away from electronics, and try a manual sync.

The watch on its stainless steel bracelet, a presentation box and the instruction manual. A paper manual is included for setting the time zone and syncing.

Is it worth buying over a cheaper quartz chronograph?

You pay for the solar movement, atomic sync and sapphire glass rather than a plain quartz chronograph. If you want a watch that sets itself and never needs a battery, those features are the reason to choose it.