Omasse is on a timer, but don't worry, it's pretty forgiving at the end.
You don't have to have Mii Plaza open to get StreetPass tags. It works like any other game with StreetPass enabled does. Just make sure your 3DS is on and that the StreetPass switch is set to 'enabled.' You'll know it is because you'll see a green bar on the top of the touch screen that says "StreetPass" on it. If it's blue it means you're connected to the Internet, which is fine, too. You can tell if you've got a StreetPass tag even when your system is closed because you'll see a green light on the top right corner come on (if it's blue you got a SpotPass tag).
In Japan it's easy to get StreetPass tags since everyone lives on top of one another there. But here in the US, where we're a bit more spread out, it's a little harder. So Nintendo did something really cool. At certain Wi-Fi spots, they set it up to save and store StreetPass tags, and you can get them by passing these Wi-Fi areas, even if there aren't any 3DSes around. Places that have these Wi-Fi spots include McDonald's, Starbucks, and surprisingly, Home Depot! Really, i was just at Home Depot last night and got 20 tags. Not because there were 20 other people in the hardware store with a 3DS, but because their Wi-Fi stations had that many stored on them!
I've helped run many a StreetPass meetup locally, so if you have any questions, feel free to ask. I know a thing or two about a thing or two.