Your starter is happiest & healthiest at controlled room temp (ie. 68-77 F) being fed on a regular schedule. If you don't want to waste the flour and water (by just feeding, discarding all but "x", re-feeding, etc.), you can find something to do with the starter (like make pancakes, waffles, scones...) for a couple of weeks before putting it in the fridge. The goal is not to see how quickly you can get your starter in the fridge, but to make sure it is lively, healthy, and active.
Once it's in that condition, feed it one more time & put it in the fridge right away. I like to take my starter out & feed it once a week whether or not I plan to use it. Starters can certainly tolerate longer in the fridge without it, but the more often they are used and fed, the happier they'll be!
Just my two cents...