I would suspect that you have the 22" model.
With that said, barrel makers like Lilja have a mulitude of custom and factory tapers in their data bases............usually if the taper is in their system, no charge. If not...........30-50 bucks one time gets it put in. Of course, a gunsmith can taper a blank to whatever you want, but can be costly because he does not want to do it....the tapering himself........time and effort is up there on a manual machine.
The parts from your existing barrel might be able to be reused...........some issues with say the band and banded front sight....but certainly the quarter rib is. One might argue about putting in the two studs for the quarter rib with 4 8-36 screws there too.
The parts you "might" need are generally available from Brownell's or Ruger themselves. The extractor is a 10 dollar part, you will need the standard 473 one, the barrel band and front banded sight are there too. Your sight insert, brass bead probably and plunger/spring are of course reusable.
I have found over the years that a simplistic slow rust blue(Laurel Mountain solution) does match pretty well the black hot blue of the No.1........in my opinion of course.
I suspect you are looking for a turn key approach, as most do not do their own work anymore. I have read a lot some on these guys from time to time for No.1's rebarreled at reasonable pricing:
Nonneman Custom Rifles
No personal referral here, just a place I see mentioned time to time.
The No.1 to be rebarreled with a blank can be pricey depending on who you use. Computer searches for No.1 rebarreling will show page after page of references to peruse and ponder. The Ruger No.1 has a lot of labor in rebarreling, as the extractor recess is clocked as is the barrel appendages. Kinda like a pre64 model 70 in labor kinda.........
fwiw