I'm not sure what zone NYC is (maybe zone 7?) but I think it is too early to prune yet. A very general basic rule of thumb is to start pruning when the forsythias bloom in your area. Usually that means that temperatures are right for the roses to start growing and it's time to prune. I tried to do the arrows picture but it was very hard to tell from your photos what was good and what wasn't so here is my Basic Pruning Primer Take off any dead wood. Any cane that is black or very dark brown is probably dead wood. Canes that are green but have a very black or dark brown streak running down the side are probably going to die soon so take them off to below the streak. However, as some canes age they can get a darker brown barky look to them and still be living wood. You can tell dead wood by looking at the center part of the cane called the pith. Start at the tip of any cane and cut off a piece about 2 inches long. Look at the center. If it is a greenish white or cream the cane is alive. If it is brown or a dark tan it is dead wood. You can easily tell the difference. The whiter center will look moist and the dark center will be dry looking. Keep cutting down the cane in small pieces until you reach cane that has a clear whitish moist center. The cane from that point down should be healthy. Once you've removed all the dead wood you can prune for shape and size if you wish. It's a good idea to take out one of any canes that cross or rub each other. Rubbing will cause damage and you could lose both canes so take one out. Take out any spindly canes with a diameter of less than a pencil. You have a lot of that spindly.twiggy side growth on your bush. Also take out any canes that grow directly into the center of the bush. You want to open up the center for good air flow. That allows the center to get more light and to dry quicker to help prevent fungal diseases. Since your bush doesn't appear to be very large you may want to hold off on thinning it too much right away though. I am not at all familiar with the method that Campanula described for getting it to put out a new cane at the bottom (basal growth) but Camp knows her stuff so I'm sure it's good advise. I was going to suggest that if possible you repot it deeper to encourage more growth. You can also then straighten it up in the pot some so you don't have to stake and tie it like that. My worry is that eventually that string is going to strangle the rose and kill it. It's wrapped very tightly around that one main cane. If it girdles the bark completely everything from there up will die. Most of all don't be afraid to make mistakes. For the most part you can't really kill a rose by pruning it. Usually it just grows back even better. And for another thing, we've all made mistakes and still do sometimes when pruning a rose. I experience that sick feeling of "OOPS, I shouldn't have cut that" usually at least once every pruning season, lol! Mistakes are how we learn sometimes and, like I said, roses are very forgiving. I know there are several rose societies in NY and I'm sure if you contact one of them they would be very happy to help you out. Or you can go on the American Rose Society web site and find a consulting rosarian for your area that you can contact. When I did that for my area the guy came right over to my house to help me and we've been good friends ever since! Rose people are really nice and love to share and talk roses with anyone interested, lol! Here is a link that might be useful: The American Rose Society

Does it get more sun a different time of day? It might be growing to get light and will want to do that again. And it looks like it is fighting with the ivy at the bottom, that might take away some of the water/fertilizer for the plant. Are those white flowers tinted pink? Kind of looks like Cecil Brunner. If it was my bush and it got some more light, I would be tempted to run wire from one metal post to the next on either side at a couple of heights and start training her horizontally to cover the fence. I would leave the last canes near the top of the fence since she already likes it there. But if it is a Cecil or Lady Banks, I would probably wait until after it blooms in the spring.

New canes coming from the base or ground are called basal canes and they are a very good thing. Let them grow and they will each have more side branches (called laterals) that will produce blooms. The more canes and laterals, the more blooms! You can prune off some of the small lateral canes that start to grow in toward the center of the plant to keep it open and airy inside. This helps the leaves dry faster and slows down fungal diseases somewhat. And if any canes grow so that they cross and rub each other you can take out one of the two and leave the other. Rubbing canes can cause damage that can let disease in too. As the plant gets older and more mature you can take out one or two of the oldest canes to allow the plant to regenerate with newer younger ones. But with such a young plant you shouldn't have to do that for many years yet.

Most roses put out a big flower show in June. If they produce flowers then and later, they are referred to as repeaters. So your roses presumably bloomed in June, and are blooming again now. It sounds like the old photos would have been taken around the late 20's. However, there is room for doubt whether or not those are the same roses that are currently there. B & W photos are nasty things to try and interpret, IME. Red roses read the same as leaves. White, pink, and yellow roses all look almost the same. Sometimes you'll get a profile view of the flower, so you can at least get an idea of how big it is, and maybe the general form, but most of the relevant information is size and general growth habit of the plant. That helps with general class, but is rarely enough to pin down a specific variety. Williamsport looks to be just cold enough that it is unlikely they are old HTs.