Our Oslo visits have been more in March than February. Oslo's winter climate is relatively mild with Central Oslo being essentially at sea level and moderated by the maritime influence.
No need to rent a car for Oslo and if anything, a car would be a "tire around the neck". Oslo is very walkable and supported by a great transit system with various transit passes and the Oslo Card includes transit. If you fly into OSL, take the train from the airport to Oslo S and the transit information center is in the Oslo S train station complex.
You can use the same tickets on all public transport in Oslo: bus,
tram, metro, train and ferry (not the Bygdøy ferry).
Where to buy tickets The Oslo Visitor Centre at Oslo Central
Station, Ruter's Customer Service Centre (in Oslo S), in most Narvesen and
7-Eleven shops, from ticket machines at metro stations etc. Single
tickets are available on the buses and ferries, but are more expensive
than tickets bought in advance.
While Oslo is at sea level, the surrounding Oslomarka is easily accessible by transit and with its higher elevation will be snow covered when Oslo has bare pavements. Ride the T1 Metro to Holmenkollen for a winter experience. https://www.visitoslo.com/en/articles/nordmarka-forest/
I usually stay on one of the budget Thon hotels near the Oslo S train station, usually the Astoria. The Thon hotels are priced with a nice breakfast. Thon hotels run from "budget" (for Norway) to luxury. https://www.thonhotels.com/our-hotels/norway/oslo/
See Rick's what to see check off list: https://www.ricksteves.com/europe/norway/oslo