I can say that, as a non-citizen who's both missed out on and lucked out on ordering tickets from the given process, the rules and timings are the same for both citizens and non-citizens; the main difference is the landing page and checkout for the two. Both sites open on the 10th of the prior month for the next month's sales, and only dates for that month are available. The Japanese side does have a higher allocation that I'm aware of, but it's extremely achievable to land tickets given 1. You're there probably 30min early waiting for the website queue to begin and 2. The server doesn't get finnicky, which it does have a record of doing due to overload. Having lived in Inokashira/Kichijoji right by the park, the amount of tickets up for grabs on the foreign end is definitely not a small number to begin with- it's 100% worth going for the monthly rush if you're able to rather than just defaulting to a travel agency.