I don't know anything about rock climbing in Hokkaido but hiking/backpacking is terrific. It can be a bit rainy in June. If you visit Lake Akan area I highly recommend hiking Mt Meakan and soaking in the onsen after.
FYI if you have visible tattoos you will not be welcome in the onsen.
It is true however, if tattoos are small enough to be covered with a bandage, you are good to go.
BTW, I went to Japan a few years ago and found few partners on rockclimbing.com. I was in Tokyo area so it may be little different but maybe it's worth a shot. Good luck! Japan is beautiful and worth a visit. (I am biased for I am Japanese. I wish I could go with you!) Oh and yes, June is really rainy there. Don't know anything about birding seasons in Japan, but there are pikas in Hokkaido :)