I have the same problem (just shoved the mattress up there and from
that experience, know that the the rigid box spring will not fit). A
co-worker suggested to me that I should cut the back slats in the back
of the box spring so that I can fold it. Once upstairs, use metal
plates to fasten the halves back together with some screws. She claims
this worked fine for over 10 years with no problems. I'm not saying
cut the whole thing in half, mind you, just the wooden slats in the
back so that it can be folded. It's up to you which way it should be
cut and folded (lengthwise vs. widthwise). It might be a good idea to
have a center support on your bedframe after cutting it like this. The
metal plates and screws can be plenty strong enough if done correctly,
but it doesn't hurt to have a little extra support too.