The RAM is only used to store the settings when powered off, so you're not losing any cal settings.
The max standby current on the RAM is 1uA @ 25C. A super-cap has leakage on the order of a magnitude greater, about 10uA for a pretty good 1F one. So accounting for a couple of Schottky diode drops to build a super-cap circuit, and assuming a 1F super-cap, it should be something around 3 days [(5.0Vsup - 2*0.2Vdiode - 2.0Vterm)*1F / 10uA]. That's not accounting for other leakage through U29C, Q1, diodes, etc., so it's going to be somewhat less time. More farads could get you more time, but there's also more leakage so you'd have to compare datasheets.
5V super-caps are not cheap. Why not just replace the battery? It's a Panasonic BR-2/3 with tabs. Here's a place that has it for USD$10:
http://store.batteryspecialists.com/br23atabs.htmlThere's also a bunch of sellers on ebay for considerably less, if you can believe their date codes.
And if the battery went bad, I would 1) make sure you get ALL the gunk off the board, and 2) check the current draw out of the battery (or temporary supply) when the unit is off. Maybe there's a reason why the battery died besides old age.