Can you get arthritis from a bee sting?

No. A person may have several severe reactions to bee stings including some types of total body pain. However, arthritis is not caused by bee stings.
Generally not. It theoretically possible to develop a type of immune reaction to a sting that could cause a type of arthritis that might last a few weeks, but I have never heard of this actually happening. It would certainly not lead to a permanent arthritis.