To get from Utoro to Rausu in winter, Navitime recommends walking. Unless you're Bear Grylls, you'd better ignore that advice.

East Hokkaido Express Bus Route 5 runs from Jan 25th to Mar 8th- It requires reservations in advance

It departs Utoro Onsen hotel at 08:15 AM and arrives at the Rausu Highway rest stop at 11:45 AM.
Coming back, it departs Rausu Highway rest stop at 4:25 PM and gets to Utoro Onsen at 6:45 PM
https://easthokkaido.com/expressbus_winter/route5/

This bus route also serves Nakashibetsu airport, so that might be your best gateway to the region in the winter.

Before Jan 25th, it's going to take you two days with an overnight stop somewhere. My preference would be to do the overnight stop in Kushiro - a long way round but relatively simple; Bus from Utoro to Shari. Then train from Shiretoko-Shari station to Kushiro. Overnight stop in Kushiro. Then a long bus trip all the way up the eastern coast to Rausu.

If you feel like having a train and bus adventure through the Hokkaido boondocks, you can take the bus from Utoro to Shari. Then train to Shibecha. Then bus to Shibetsu *(or Tawarabashi - but unless you have to, I'd avoid trying to be too clever with connections in isolated places like Tawarabashi). Then bus to Rausu. I leave you to work out the times and the accommodation options. If you're serious about doing it, you should call ahead and verify that all the services are running on time.

Winter bus timetable from Utoro to Shari:
http://www.sharibus.co.jp/pdf/2019_siretoko.pdf

Train times from Shiretoko Shari station to Kushiro (via Shibecha) are on Hyperdia. (Type "Shiretoko Shari" to get the station in Shari)

Bus info. from Shibecha (via Tawarabashi) to Shibetsu is here:
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Bus info. from Kushiro (via Tawarabashi and Shibetsu) to Rausu is here:
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