Ban an import that is exported Norway intends to shutdown the fur farming industry over the next 5+ years. In the meantime, can Norway ban imports and sales of fur without violating WTO regulation?

Are there alternative strategies? Examples of countries doing something similar (banning imports of goods it exports)?
In general, no, it can't impose stricter rules on imports than it does on domestic production. Or more accurately, it could, but the countries exporting fur to Norway could challenge this. The guiding rule is that domestic industry cannot receive preferential treatment over importers.

Norway can do it the other way around - that is to say, WTO rules would allow it to continue to import furs after closing down its domestic industry.

But there are always way around this. For example, by setting very specific standards (which are not explicitly to do with country of origin) that apply to everyone, but that only domestic industry complies with. If the standards are crafted well enough, they'll pass WTO's scrutiny.