New Japan
NJPW does not make it at all easy to buy tickets overseas. There are several ways to do this, and there is roughly a direct relationship between effort required and chances of getting a ticket, and an inverse relationship between those two and realistic prospects of trying the method in question. Note that this outlines my thoughts and experiences for buying Fantasticamania tickets for Korakuen Hall, which seemed like a near-worst-case scenario in terms of selling out fast. Other events (like Wrestle Kingdom, which runs in the much more spacious Tokyo Dome) may have different ticket dynamics.
In roughly increasing order of convenience for gaijin:
Fan club
If you have a friend in Japan who is willing to lend you a Japan mailing address to receive stuff on your behalf, and you're willing to invest some cash for the membership fee, it's possible to sign up for the NJPW fan club, which often gets first crack at tickets (but even then, sometimes there is a lottery system). I wasn't willing to burden someone (or my company's Tokyo office) this much for a better chance at tickets, so I passed.
Buying online
New Japan has a few different ticket partners. Ticket Pia is the general one, I believe, but sometimes a different one will be used for the presale. To register for Ticket Pia, you need a Japan phone number that you can dial out of on short notice; this is how they verify that you have a working Japan phone. You can get a Japan SIM card in advance of your trip, but you won't have international roaming access, so that doesn't help for this purpose. It's possible to get a DID (direct inward dial) number through virtual phone number providers (I used FlyNumber, but there are others), and set up a phone dialing app to call out of that number. Due to telephone regulations in Japan, you will need to scan IDs and provide certain other information in order to get a phone number, but this wasn't too onerous. I didn't end up buying tickets through here, but I am registered.
For Fantasticamania 2018, NJPW ended up using Lawson's for the public presale. You need to give a Japan phone number and mailing address, but as far as I can tell, neither is verified. Buying tickets seems restricted by IP address (the error message is a bit confusing, to boot), so you may need to VPN into a Japanese IP to get this to work. When you buy the tickets, you will have the option to settle now with a credit card, or to settle when you pick up the tickets. You must pay in advance with a credit card, or else the reservation will be canceled for nonpayment within a few days.
When you get to Japan, you can claim the tickets at a Loppi machine at any Lawson's. Basic knowledge of Japanese helps here, although you can also print out comprehensible Google-translated instructions in English on Lawson's website in advance. You will need the reservation number and the phone number that you registered with. Once you print the claim slip, you have 30 minutes to check out at the register to have the actual tickets printed.
As far as availability goes, the second night sold out during the presale, but I was able to get decent seats for the other two nights online by getting up early when they went on sale. It definitely wasn't a "PWG BOLA sellout in 90 seconds" situation, but I wouldn't push my luck.
Ticket-buying service
There are third-party services that will, for a fee, attempt to buy tickets for you, bypassing all of the crazy things like registering for a phone number. I surmised that I'd have a better chance if I went the DIY route with the presale since I'd be in control over what time I bought tickets. However, I do know of people who have used these services with success, so it's an option.
Walk-up
If all else fails, there will be standing room tickets available the day of the show, at least for Korakuen Hall. Note that fans will reserve spots in line for themselves early by taping newspaper to the ground and writing the current date and something along the lines of "新日立見" (New Japan standing room), and this is deemed acceptable and is honored. Barring this, you'll want to show up to the ticket booth line (which for Korakuen Hall starts along the side of the building) at least a couple of hours early. Once you get the tickets, you'll need to line up in the standing room entrance line (in Korakuen Hall, this is the stairwell leading up to the fifth floor of the building). Again, this is first-come, first-served, and people are smart enough to tape line placeholder signs to the wall.
For what it's worth, I didn't do the placeholder signs, but showed up just less than two hours before the ticket booth opened at 4pm for the second Fantasticamania show, and got tickets.
Dragon Gate
Send email (in English) to the email address listed in @DragonGateEN's Twitter account to reserve tickets. You can pick them up and pay for them in cash at the venue just before the show starts (for Korakuen Hall, the pickup location was the 5th floor entrance).
For what it's worth, when I went to the show, there were still reserved seats available when the ticket booth opened for walk-up sales, but you'll obviously be able to get better seats by reserving in advance.
From what I saw, other puroresu promotions running Korakuen Hall had walk-up tickets still available, so I'd assume that NJPW is the only puroresu case that you'd really have to worry about for Korakuen. Unfortunately, it's a pretty significant case.
TL;DR: Start planning early if you want to buy NJPW tickets before you get to Japan and don't know anyone already there. You will probably need a Japanese phone number, a way to VPN into Japan, and quite a bit of patience with the ticket website registration processes.