This is pretty similar to how I root cuttings too. A few differences, but close. At least close enough that maybe I can help about how often to add water.
I just pour water onto the medium, and I don't mist the tops. When the air is dry (e.g. winter indoors with forced hot air heating) I use bags over the tops rather than in a bin. And when the air is more humid (the northeast is generally humid, after the furnace is done in the spring), then I just leave them open on top. I never use bins... not against it (and it seems like a convenient idea... I've just never done that).
As for how often to water the gritty medium, I just want them to stay moist, not dripping. As a rule of thumb I check every three or four days... if it's getting dry, I add water (it drips out so quickly that I don't really worry if I've added too much all at once... it drains quickly). It seems four days is about how often for the dry indoor air I've got, but sometimes it's longer. I look at the insides of the cups for condensation... if it's there I figure they're still moist. I do two other things to monitor moisture. One is pretty obvious: I touch the perlite on the top (sometimes with a tissue and sometimes just my bare finger... the tissue helps me know if it's moist or not). The second is a little trick I came across by accident: I put a couple of pebbles (ordinary bluestone gravel, small bits of roundstone or creek gravel, or even slate/shale) in the bottom of my clear cups before I add the gritty mix and the cutting. The point being that I can see if the pebble looks moist from the color of it.
I have other differences too: my most-frequently used mix is mostly perlite (about 80%) with just a little bit of promix or other soil (usually a soil with some amount of organic matter). The soil/organic matter is to provide just a small amount of nutrient once the roots do get started, and also to hold moisture). One other thing I do is I cut holes in the sides of my cups to promote gas exchange (that helps to evacuate the CO2 that is formed by root formation... the CO2 is a byproduct of root formation, which, if it collects too much, will inhibit the newly forming roots from continuing to grow... suffocates them). Most fig varieties don't seem to need all this extra attention (lots of them root like weeds... and rooted just fine in sand+peatmoss plus being ignored)... but I guess they don't mind having good conditions either.
Good luck... I'm interested to see how it works out for you.
Mike central NY state, zone 5