I'm slowly buying parts to build up a new bike. I just bought a JIS Hatta Swan headset and afterwards realized that I didn't check the stack height. After some searching (it's not listed many places online) I found the stack height to be 36 mm. I was going to put it on a Alien Frame (which I haven't bought yet) which their website says has a 32 mm stack height. Is the Hatta Swan going to work? As long as it grabs a few threads I'm okay with it not being perfect.

No. If the fork steer tube is only 32mm longer than the head tube, you're at least 4mm short. One or two millimeters you can sometimes get away with by leaving out a washer and only engaging a couple threads on the lock nut, but 4mm is almost certainly too much to cobble around.

Whoops. Live and learn, Newb. Why don't you send that headset over here to someone who knows what to do with it?