OP, when playing the higher difficulty levels, playing a specific strategy is FAR more important than micromanaging things. You can switch things up, but you have to have a strong opening plan at least. If you're battle oriented, having a good grasp on combat strategy is really key. As many have said before, you should be able to take on much stronger AI armies with your keen human mind!
Regarding micro, I usually just set my cities on "default" or "food" early on. Then after my cities get up to size I tend to switch most of them over to "production" focus. The only time I micromanage things is if I *really* want production focus, but there's not enough food to support it. Under those circumstances I may temporarily set to "production" and then switch out a few population points to optimize. But that happens very rarely.
Whenever a city produces something, that's my cue to check on things quickly. If a city is producing things quickly, then I won't bother checking in for a dozen turns or more.
Also, don't be afraid of the "avoid growth" check box for some cities. On easy difficulties, happiness isn't much of a problem, but on higher settings, getting into a happiness hole is dangerous. If you're in one, it can be hard to get out, so you need to avoid it as much as possible. Though keep in mind, that in some games I'm straddling the -9 happiness barrier quite often!
It's not so much about micromanaging cities, as it is knowing how each city will be used based on the terrain. Let it do it's thing, but guide it a bit from time to time.