Agree with Alice, you can grow any roses from cuttings, grafting is just a way for growers to propagate large numbers quickly as one bud is all that's needed from the parent/grafted plant whereas cuttings usually are taken as 12" sections. The same applies to fruit trees with some exceptions.
It might be worth identifying the rose to get an idea of it's natural habit and see if that's what you want - it sounds like you want a climber/rambler. Also can you see that the original plant is grafted? I suppose these days they all will be but it might be from a cutting itself?
In any case grafting isn't difficult, mid summer is the time, they say to do it in the afternoon so the cooler evening is easier for the plant to adapt to, but in this weather just keep it in a cool place. The graft should heal in a few days and show signs of swelling and growth after a week or so.
Dr Huey is a good choice, it is very vigorous and tolerant of a variety of conditions - as a rose in it's own right it's very susceptible to fungal disease, has little scent and doesn't repeat flower so not a great garden choice but none of these problems are passed on, only the extreme hardiness!
If you do just fancy cuttings it's better to do them in late summer/autumn as hardwood cuttings. Just take a 12" piece of this year's growth and bury it about halfway (right way up) in a quiet corner of the garden. It'll sit there happily til next summer or if you can wait til the autumn better to move it then.
HTH
Helen