If one rebores, then the existing barrel, sights, finish, and markings are pretty much retained. Reboring and chambering can run north of $250 if the rebore is feasible from their point of view. Ruger drills/taps the quarter rib holes very deep on the No.1(especially the forward ones), so that comes into consideration. After the rebore, the chambering has to be done or at least the neck. I would certainly touch base with the reboring facility of choice on feasiblity and services.
I personally would favor a new barrel blank. In order to keep the No.1A and I am assuming the same AH forend without mods to said forend, then you might search for a barrel maker that will do the specific No.1A taper on his offerings. Lilja offers such in his data base, I suspect that PacNor does as well. Only two of many that has the Ruger tapers in their data bases. I was not aware that Shilen would custom taper to a No.1 contour, but of course you could inquire. A Shilen #1 taper might suffice, I really cannot say from here.
If you plan to retain the iron sights etc of the old barrel, then they can be fitted to the new one. The quarter rib etc should be able to be saved with not too many issues, perhaps new screws and alignment studs. Ruger has a thing with adhesives on these pieces. The front banded sight is listed by Ruger and by Brownells as a piece part, but I would inquire on availability first, as these are funny times right now. If you do procure a new No.1A factory front banded sight, it will have to be fitted or gaps will show as the piece part is straight bored and the barrel has a taper. If you find a patient gunsmith that will bore out the existing front sight from the old barrel, it may be pricey as tis time on the lathe, plus the hassle of doing that, over the 30-50 bucks for a new front banded ramp. A lot will not place into the lathe for that.
Nothing is magical about rebarreling a No.1, anyone that does basic gunsmithing can do it. The rebarrel will be more pricey than say a Remington 700 rebarrel because of the ejector cuts, and 4-6 holes in the barrel; then the front sight fitments. No.1s in a rebarrel are more time consuming than due to things that have to be timed in angle; hence the added costs in install.