I am Japanese and yes, to be on the safe side, I would plan for it to be a problem and not just at Disneyland. Very few people will assume you are yazuka/dangerous, but there is a deep-seeded cultural bias against large tattoos, especially sleeve/full-body tattoos. A butterfly on the ankle, a tiger on the bicep, not that big of a deal. But a sleeve will definitely draw undesired reactions. I know that others have gone to Japan with tattoos and have had few problems, but I cannot emphasize this enough - a full sleeve is judged very differently than a large tattoo.
Be prepared to be turned away from some businesses if you show your tattoo - spas (even high end ones catering to tourists), onsens, ryokans (even in Tokyo), even stores/restaurants in very rural areas, and all pools.
For the record, I am not anti-tattoo so I don't want to sound harsh. I've got a medium sized one on my back that I can easily hide when I'm in Japan.
And the good news is that if you cover it up, you'll be ok. I would suggest long-sleeve moisture-wicking shirts. Depending on what part of June you're there, the weather is pleasant and I would usually still be wearing light sweaters at that time of the year. It's August that REALLY sucks.
The other good news is that people in Japan are VERY polite. They'll do their best to ignore your tattoo as long as you make an effort to hide it. You'll gain a lot of "face" by doing your best to try and respect the culture.