The only reason I gave you that link is because I know the seller to be honest and reliable ...
During its entire period of production, (1877 to 1909) Colt made a point in later years to point out that it wasn't certified to use smokeless powder, just black. They went so far as pasting a warning label on the box to this effect. So to start with you are limited to using black powder.
The .38 is a better choice as a shooter (if there is such a thing as a "better choice") because the cylinder walls are thicker then on the .41, but not by much.
The .38 Long Colt cartridge originally used a heeled bullet, like today's .22 L.R. The front part of the bullet was the same diameter as the case (.375) and the back, which fitted inside the case was .356" to .358". I'm not sure if heeled bullets are available commercially, but bullet molds are - or at least were.
You can make brass by trimming .38 Special cases, (or for that matter use untrimmed casses - the chambers are bored straight through - although I understand that some of the smaller ammunition makers that supply cowboy action shooters are offering .38 L.C. - but it is loaded with smokeless powder so don't consider using it! I believe the new stuff is also loaded with regular .38 bullets sized to .358" - the undersized bullet would shoot O.K., but decent accuracy would be a something thing, maybe good but likely not.
At one time, .38 L.C. was offered with .358" diameter bullets with a hollow base. The idea was that the base would expand and seal in the oversized bore. Again, maybe, maybe not ...
You could try loading a .38 L.C. case, or trimmed .38 Special with a 148 grain full wadcutter with a hollow base. These are commercially available. The weight of the bullet is pretty close because the original loads used a LRN bullet weighing between 146 to 150 grains. Seat the bullet to the mouth of the case, and if it doesn't want to go all of the way in, reduce the powder charge. With the bullet seated, the powder should be slightly compressed.
I have shot model 1877's on a VERY LIMITED basis, but don't recommend it. If something breaks in the lockwork you could be in a real fix trying to get it repaired. Broken lockwork relates more to how many times the action is cycled, rather then actual shooting.
I understand your desire to shoot this interesting and very historical revolver, but I hope that your experience is limited. The remaining guns are getting very expensive, and are not particularly good shooters from an accuracy standpoint.
Edited to add: Apparently the revolver is in reasonably good shape. One grip is cracked, but original style replacements are available for about $20.00 or so. It undoubtedly needs a good cleaning and lubrication, with come extra attention to the bore and chambers. Beyond that I think it is shootable (on a very limited basis) within the qualifications that have been discussed on this thread - in particular, using black powder only. You also have to understand that the gun is probably over a century old (maybe well over) and subject to small part breakage. Shooting original antiques carries some pleasure, but a lot of risks.