Hi, I'm Japanese and I applied Outland and was approved in 2014 so I might be able to offer some pointers.
The only thing CBSA is interested in when your girlfriend arrives in Canada is if she understands the conditions of her stay (that she cannot work or study, that she cannot stay indefinitely, and that if she wants to stay in Canada long-term, she needs to apply for PR). To demonstrate that, a return ticket would be helpful, though not mandatory. If you are applying for PR immediately after marriage, you could pay the fees for her PR application in advance and bring the receipt. That will help demonstrate that she is trying to stay in Canada legally and understands the rules.
My husband and I were married in Japan (no wedding ceremony, just submitted documents at city hall) and we applied immediately after the marriage. It took city hall about a week or 10 days to update my koseki so that I can get a new updated koseki tohon with my new marital status. I think I've heard that in Canada it may take a bit longer to get the marriage certificate.
For the proof that you are legally free to marry, she should make an appointment at the Canadian consulate in Japan, (don't do it at the last moment, because if you can't get an appointment, you can't get married) and when you are in Japan, you go to your appointment, meet the notary and sign an affidavit saying that you are single and able to marry.
As Rob_TO said, as far as immigration is concerned, engagement is irrelevant. The only ways for you to sponsor her for PR would be to marry her, or cohabit for a minimum of 12 consecutive months.
I came to Canada after submitting my PR application. As I was already married and my husband was in Canada, I thought CBSA might be a bit skeptical about my 5-months-long "visit". I came prepared with proof of my PR application, the email saying my husband was approved as my sponsor (stage 1 approval), a return ticket (which I refunded the next day), over 10,000 CAD for my stay and two very large suitcases. At passport control the officer barely looked at me and asked me less than 5 questions. She asked when I'm leaving and I gave the date of my return ticket, she asked where my husband works, and where he lives. She didn't ask if I've applied for PR (she may have had that info on her screen though, I don't know.) and that was it.
So it's best to be prepared for the worst and hope for the best. Most people don't have any problem getting in. The best thing is to NOT lie, be open and honest, but don't offer more information than asked. Just answer questions truthfully. (It would Not be a great idea for her to vaguely say "visiting some friends" - she should just say she's coming to visit you, or to marry you. When I visited my then-boyfriend the officer seemed very suspicious when I vaguely said "visiting some friends, going kayaking" and while I luckily did not get sent to secondary for questioning, I sweated a lot while she asked a few more questions...)
And you may not have read up much about Inland/Outland, but here's my 2 cents anyway. As a Japanese citizen she should apply Outland. It's way faster than Inland and doesn't restrict her movements. (Inland requires her to stay in Canada for the entire duration of processing, no matter how many years it takes.) She can physically be in Canada when she applies Outland. She can give a Canadian mailing address and still apply Outland. A lot of people apply Outland and still wait it out in Canada, like I did. She can't work until she gets PR, but it'll only be a few months.
Ask away if you have any questions!
