There may be a capacitor, but it's probably more likely that it just uses a trickle-charge to keep the save chip running while the system is plugged in, so you'll only learn your battery is dead if you unplug the system or have a power outage.
In comparison, the Dreamcasts' clock battery only charges when the system is on, and many have died -- I need to set the clock pretty much every time I turn the system on, annoyingly. I know it's rechargeable, but there is obviously a limit to how long it lasts...
If you mean the Memory Base 128 (or Save-Kun), what you need is one of the games with a built-in memory manager that lets you copy banks of memory between the MB128 and the system -- there is no built-in manager, so only games designed to support the MB128 can see it. I use Emerald Dragon's manager, since it's simple and easy to use (hold Up when you hit Run to start t the game from the CD system menu). The batteries should last at least six months per set. The MB128 also does have a capacitor in it, but it won't last long at all when the batteries run out -- maybe an hour, not overnight even. It will draw power from the system to keep files as long as you keep the system on even if the batteries are removed, though, which is nice.
I replaced the clock battery in my DC, but it seems like I've never had it on long enough to recharge it. It now acts as dead as my old one. Bummer.
I have the Memory Base 128, I think. I actually have a copy of Emerald Dragon too! I bought it for like $5 ages ago, and it's just been sitting on my shelf. I should connect everything and give it a try.