You can do it. It's really not hard, certainly for a component like this.
Quick check though-- what would you do if you didn't try to solder it. Before we encourage you to do something that might destroy the board (probably won't, but mistakes happen), it would be good to know the loss wouldn't be too big...
As far as what it does, it's probably just part of the power supply that keeps enough charge close to where it's used so if the circuit suddenly draws power it can take it from that capacitor in less time than it takes for it to make it from the main power supply. It's called a "bypass" capacitor, or filter capacitor because it removes the high frequency current demand. The symptoms you're seeing would be consistent with that-- works fine some times, crashes for no apparent reason (the unapparent reason is a sudden demand for power that causes a brownout without that cap in place).
That square black chip with all the pins is a Linear Technology 3731 voltage converter. I can't be sure without testing the board, but I'm
guessing this is the 10µF on the Vcc line. Vcc is on the side of the chip closest to that cluster of parts. You want that part there, it's probably not going to work to limp by without it.
In the end, the point is to get the ends of that part electrically connected to the board. That's all there is to it. Better tools make it easier, but this isn't anything too fancy so the stakes are lower. You could add solder to ensure a good connection, but you might be able to just reflow the existing solder if it's not worth buying more. If you add solder, don't over do it.
Things to be careful of-- make sure the board isn't connected to anything when you solder it. If the tip is active, as
@NikolaPPC warned, or grounded as is common with other irons, you could push current through a board if it's connected to anything. The other thing is don't just look at the component you're soldering, but the ones around it-- you want to make the connections you're trying to without making any other ones... ;-)
You want enough heat to heat the solder, but not so much you that you lift the copper pads off the board.