1.5 day would be the minimum. See the estimates below.
I assume a day from Aug 9 to 17 here.
http://www.sharibus.co.jp/pdf/2017-04-P1.pdf8:30 A bus from Utoro to Shiretoko Nature Center (SNC) (10 min, 40 min interval)
8:40 SNC (average duration of the stay 30min)
https://www.jalan.net/kankou/spt_01545cc3290031450/9:10 A walk to Furepe no Taki (20 min to get to, 1h stay)
https://www.jalan.net/kankou/spt_01545ab2040005364/10:30 A walk back to SNC
10:50 A bus from SNC to Shiretoko Five Lakes (SFL) (15 min, 20 min interval)
11:05 SFL (2h stay and 1h lunch)
https://www.jalan.net/kankou/spt_01545ab2030004800/13:10 A bus to Kamuiwakka Yu no Taki (KYT) (28min, 20 min interval)
13:38 Kamuiwakka Yu no Taki (Onsen hot water river, assume 1h40min similar to Furepe no Taki)
15:30 A bus back to Utoro (15:58 SFL 16:10-16:25 SNC 16:30-16:40 Utoro)
They do not provide details on Shiretoko Shuttle Bus (SSB). But, you would need a full day to do the above. SSB runs also from Aug 1 to 25, with lesser frequencies, 40 min intervals in all the segments. You would need more careful planning for the bus uses. After Aug 25, only Shiretoko Line runs. Only six buses a day. No bus to KYT, just up to SFL.
Rausu line goes to Shiretoko Pass, Rausu Lake, and Rausu. Only four buses a day. If you want to attend a cruise on the Rausu side and/or go trekking to Rausu Lake, you should need another day.
The tour bus Fureai-Go B Course includes Shiretoko Pass. The series of Fureai-Go A, B, and C courses cover Oshinkoshin no Taki, Shiretoko Pass, Shiretoko Five Lakes, Shiretoko Nature Center, and Ten ni Tsuduku Michi (The path to the Heaven). These could be done in a day from 8:40 at Shari to 15:00 at Shari, not from/to Utoro.
You may add Oshinkoshin Falls after the Five Lakes sequence above: 17:00 Utoro 17:10 Oshinkoshin Falls (30 min) 17:58-18:10.
https://www.jalan.net/kankou/spt_01545ab2050006459/And you would be likely to board on some cruise at the Utoro side. A cruise to Cape Shiretoko would take 3h45min. There are other shorter cruises targeting to see animals. There are still other activities, sea kayak, cycling, helicopter cruising, and so forth.
https://www.ms-aurora.com/shiretoko/images/leaflet.pdfSo, I could account for one day for Five Lakes, half a day for a cruise, and one day for Rausu. Three days would be ideal. Two days of dropping some. One day just some pickups.