I have the same problem on both of my wheels. The rust isn't horrible but it's beyond trying to clean up. Also, when I got the tire and the tube off I discovered an unacceptable amount of rust on the inside of the wheel.
I had planned on painting mine from the get go but the rust makes it a no brainer. And yes you can paint chrome. I havn't read up on it yet but I think it involves a self etching primer. Mine will get a good going over with the wire wheel, then some sanding , followed by a rust converter, followed by the best technique for painting chrome. This has been done alot by others and the results look nice assuming you don't have to have chrome.
Lastly, some of the spoke ends on the inside of the wheel had an unacceptable amount of rust so we get a new set! Whoopee!