Good news: this process has not been as onerous as anticipated. After 10 days, I've been able to list all the country names and all but 5 of the capitals (all in Africa, because a) woefully ignorant of the geography and b) Africa is huge). I want to see if I can repeat the countries for a couple days before I check the continents off on my list, and I don't know what I'll need to do for long-term maintenance of the information, but it's definitely progress. After I've got the capitals down, I'll move on to map locations.
Initially, the only continent I could fill in all the countries in was South America. This surprised me, other than it has the smallest number, because after Africa, South America is probably the continent I know the least about. Since I have a fairly spatial memory, the continents where I do know where or what region a country is located were easier for me to ramp up quickly. Once I start working on the map part, it will probably help with long-term retention.
If this is something you'd like to try for yourself, my process has been pretty simple. First, I made a spreadsheet with a sheet for each continent and made lists of all the countries and their capitals. Then I made a worksheet on another tab to print and fill in, and I made a sheet to track progress in each category. I filled in what I could for each continent, then checked it against the spreadsheet lists, filling in what I missed in a different color. Writing by hand helps me learn things, so repeating this process daily has been useful.