While the main roads in Norway are well maintained in winter and easily navigable for anyone comfortable driving, the experience can be very different than regular summer driving. There are several issues here:
- Most main roads outside Eastern Norway are two-lane only, without any barrier separating traffic in opposite directions. You can examine road conditions in general using Google Street View, which has relatively good coverage in Norway.
- Heavy snow can make driving difficult, and in some cases dangerous, as it cannot be cleared fast enough
- Some roads close permanently in winter. This does not apply to your proposed route across Vikafjellet, though according to this news article (in Norwegian) it is frequently closed temporarily because of bad weather; in the 2013/2014 winter (apparently a bad season) it was closed for more than 1000 hours total. New Year's is in the middle of winter and the road would have a certain risk of being closed.
- If you are uncomfortable driving it might be easier to drive during the (limited) daylight hours.
Alternative routes
A less weather-prone route would be to go north on the E39 toward Sognefjorden, cross the fjord by ferry, drive east, and then cross the fjord again by ferry. This route keeps a much lower altitude and I do not think any part of it has substantial risk of being closed save in the most extreme weather.