Can I increase my power output from my solar panel using a capacitor?
You can't get power out of nowhere, no matter what you do. So no way you can increase power. Period.
Charging time of the capacitor is 5T = 5RC. It comes from exponential equation, and after 5RC you have 99% charge, usually considered full charge (or discharge, it's symmetric). I guess you aren't exactly familiar with electronics, you should read a few things about how capacitors and resistors work, as well as Ohm's Law and power. You can also find some really friendly vids on the youtube. It's all information I can give you regarding your questions. Check out those vids or articles, it will cost you half an hour time, but you will already be able to answer your own questions, it's pretty basic stuff you're asking.