Trim the leads of the new capacitor so that they are both even, and will sit at about the same height as the old capacitor. Position the new capacitor leads at the holes where the old capacitor was, with the correct polarity. Just like before, press the tip of the soldering iron directly onto the joint in the back of the circuit board.
That subtle "popping-up" is exactly what you're looking for. Make a note of the polarity of the old capacitor, and mark the exact values you'll need for the replacement: capacitance and voltage/temperature ratings (these may be written on the part itself, or you can look up the part number).
The easiest way to find out what replacement capacitors to buy is to find the original capacitor datasheet. Sometimes you can find capacitor datasheets by going to a company website. You can find some capacitor company websites here The badcaps forum is also a good place to find information about capacitors.
Press the tip of a heated soldering iron directly onto the solder joint on the back of the circuit board that is holding the old capacitor down. Hold on to the capacitor itself with your other hand. As the joint melts, you can feel the tip of the iron fall into the hole of the circuit board.
I remove the two capacitor clamp screws. Be very careful not to slip with the screwdriver as a scratch on the coil will be fatal to the watch. This allows me to remove the clamp, insulator and capacitor, ready for fitting the replacement capacitor. Be sure to make a mental note of the contact points as you remove the old capacitor.
First, go to the website of your electric components distributor and go to the Aluminum Electrolytic Capacitors section. Narrow the search by entering the capacitance (uF) and voltage (V) values of the old capacitor. You may also want to check the box to only show components that are in stock. Narrow the search by price.