So I've just acquired a 2003 Montero XLS and have noticed this strange flapping noise coming from the engine, audible with the hood closed. My parents bought it new but have since replaced it with something more fuel efficient. I dont remember if ever sounding like this, at least not 10 years ago. Wondering if its something I should be worried about, or just the way they sound after +170k miles? I'm ordering a bunch of parts to do some work on it and if its being caused by something, might as well take care of it while I'm doing everything else (spark plugs, valve cover gaskets, timing belt/water pump).

It usually takes a bit longer to start than this, but here is a video of the

 

Two possibilities:
1. You need to bleed valve lash adjusters.
2. your timing belt tensioner is bad.
Does the sound goes away or "improves" when you rev up a little? What kind of oil are you running in it and have it run low any time recently?
One thing for sure, looking at your video you will need a new serpentine belt tensioner. It is flopping all around and it's not suppose to do it so much.
Word of advice - if your water pump is not leaking - leave it alone. There is no reason to replace it if it is in good condition. Mine started to seep just a little fluid out of the seep hole at 250k (original pump). I finally replaced it and out of curiosity opened up the case to examine it. Internals on the OEM pump look pristine. No rust, no wear on the impeller, no clogging. The quality of that pump is light years better than anything you'll replace it with and replacing it before it is actually worn out is equivalent to throwing money away.