No jokes, seriously.

My 'e' key has been sticking lately, and it gets annoying when I try to press it and it just sticks.

Any way to fix this without getting a new keyboard?

NO ♥♥♥♥ JOKES, DAMMIT.

Take off the E key, clean it out, put it back on.

Laptop or desktop?

Desktop.

And is it just as easy as popping it out and back in?

Ideally you should take all the key off your keyboard with a spoon handle or something, and just run the damn thing under a tap, then let it dry for a few days, and run a hairdryer over it.

In practice, sometimes the running underwater part breaks some keyboards, and of course you need to have a photograph of your keyboard from before so you know where to stick the keys back on.

Alternatively take off the E and wipe the area with a damp cloth.

I'm going to try the alternative of just wiping it off.

I just don't want to take the key off only to find out it won't go back on, so I'm going to ask out of paranoia: Do all keyboards have keys that pop on and off?

With all the varied keyboard models?

I doubt it, but it's pretty common for the aforementioned purpose of cleaning them.