From the size of the rose in your photo, I think you need to find a larger pot, if possible. I would probably go to a nursery and get a 20 gal nursery pot that they use to grow small trees in for this rose. Rose friends of mine that live in LV paint them white because they have told me the white reflects heat and keeps the roots cooler. I haven't done that myself, but I thought I would just pass that information along.
In a sense, roses are like trees. The root mass under ground is as large as the top growth above ground. This why you want to have as much room for the roots as possible when you are planting a larger rose.
Also, you want to place something like pieces of tile under the edges of the pot so that it doesn't sit flush with the hardscape. This will help with drainage. You don't need to spend the extra bucks for "pot feet".
I hate to say this, but from the photos, your rose is terribly stressed. Roses will abandon the top growth in order to survive. Your rose has not really put on any significant top growth. It's in survival mode.
As to the order to do things ... chop the top growth back before you pull it out of the old pot. It will be easier to handle the plant when you do the root prune and re-pot the rose. The next step would be to do the root prune. I am guessing that when you pull the plant from the old pot you are going to see all roots and no soil. The goal is to open up the root mass so that when you plant the rose you can get new soil all around the roots. If you have seen a bare root rose, you will see that they have removed almost all of the roots. The roses do come back.
When you plant your rose, it's going to look like it is just sitting there doing nothing and you are going to think you have killed your rose. Actually, the rose is doing what roses do ... it's growing it's roots first. You just can't see the rose doing what it needs to do to get ready to support top growth. This is where you will have to be patient. Just water it the plant. You want to keep it moist and don't let it dry out. You don't want it to be too wet ... except when you do a deep watering and then allow it to dry out to moist. If you keep it too wet, you will end up with root rot.
If you grow house plants, I am certain you have a feel for this. If you have more questions, please feel free to ask.
