We had access to a car for this itinerary, although I'm sure it would be possible without one; perhaps with a bit more planning though. Pack a picnic lunch to enjoy in the park. If not, there are plenty of dining options around, including a café in Vigeland Park.
The Royal Palace gardens are only open in the summer, but a visit to the Royal Palace is a must-do, whatever the season.
Norway is an expensive country, so be prepared. Save money on things like meals by packing a picnic lunch, in order to enjoy the other sights without worry. Also, if you are planning to visit a few of the museums in Bygdøy, I believe a museum pass is available to save money.