I suggest building the arbor, cutting the rose way back and moving it to the new location. As soon as new growth comes, you can tie it on there loosely, leaving room for canes to grow.
If the rose is grafted, don't cut it below the graft.
Always leave some leaves on your rose when cutting back if it's not dormant!
I don't feed mine directly after transplant. I'd wait to feed again until late July and give it one feeding and no more. Let the roots get adapted, and you don't want too much luxuriant growth going into late season. Just enough to get some fall bloom!

GGG -- actually, we are planning to leave the rose where it is, and build the arbor under it. Do you think we should still cut it back? Or just sort of push the long stalks back while we build, then let them fall back onto the structure?

Try just tying it on, if that won't work, prune to a good height for tying. Roses are forgiving. VERY forgiving!

If you can build the arbor around the rose, then keep the canes. Then tie the canes to the arbor, don't weave them in and out of the arbor, that damages the canes. Your goal is to get the canes to be horizontal when they cross the top of the arbor. The rose will then shoot laterals up and those are where the flowers will come.

If you wind up having to cut the canes, don't worry. You really only need two or three big canes, and as GG said, roses are very forgiving and will grow if cut back.