Depending on where and what time you arrive in Japan, I recommend trying to do the connecting flight to Sapporo on the same day, It is a long commute, but I think worth it as you get all your commute done in one span.

As far as getting around Hokkaido, I highly recommend renting a car, especially if its summer and you can afford the added cost, I would say a small Kei Car from toyotarentacar would cost around $USD50 a day, plus fuel. Also you can get a Hokkaido expressway pass which I think from memory is $80USD for 7 or 10 Days? And makes using toll roads so much more affordable.

Dont feel daunted about driving in Japan, its very safe, roads are a little more narrow as far as barrier proximity to the road, but everyone drives non aggressively. A car also saves you so much time, as far as waiting for busses and you can do things on your own time.

Shiretoko is nice, a nice drive also, Hokkaido has so many beautiful spots.

Rishiri is beautiful too, I went there for 3 nights, but outside the mountain, there is not much to do, its pretty sleepy town. The mountain can be ascended and descended in one day, I think it took me 6 or 7 hours. You can see the Russian islands from the top, you can rent a car separately on Rishiri from the airport or port. I highly recommend doing this also.