...You can shoot modern ammo (if is the rifle is rated for smokeless) but those early 38-55 were all generally all .380 groove dia. Most modern 38-55 rounds are loaded with .375 dia. slug so your accuracy will not be very good at all.......
I think you'll find that it's the other way around from what Muskyhunter says in his post.
.38-55 went through a shift in size during it's life. The two single shot guns I've got in this caliber were slugged and found to want a .376 to .377 cast bullet to fit the groove diameter correctly.
At some later point in history when the .38-55 came out in lever guns the bore and groove diameter grew to where these guns wanted a .379 to .380 size bullet.
There's also two different lengths of .38-55 brass to choose from. The original longer style and a slightly later shorter version. And with your 1885 being an original and not a later reproduction there is a good chance that you will need the longer original style casings.
So all in all you might be able to try some modern factory ammo. But I suspect you'll need to make the jump and get yourself into reloading if you expect to see the best accuracy out of this fine looking rifle.
Two things that you will likely need to do or have done is to slug the bore and cast the chamber.
Slugging the bore is easy enough. Take a soft lead round ball from a muzzle loader and pound it into the barrel at the muzzle with a plastic mallet. Once it's mostly in use a 3/8 dowel to punch the slug down and out at the chamber. Now you can measure the bore and groove diameters and get a better idea of the bullet diameter you need.
Casting the chamber is a bit harder. You want some sort of low temperature metal like Cerrosafe or to plug and fill with a casting product like possibly plaster of paris or Ultracal casting mix to make a plug of the chamber and the first 1/2 inch of the rifling. The resulting plug will allow you to get an idea of if you need the long or short version of the cases.
I know this sounds bad. And it is thanks to the changes that .38-55 went through over the course of this cartridge's life time. But if you take this on and accept the work needed you'll have a great shooting rifle in a cartridge that doesn't beat you up.
.45-70 is a hugely popular cartridge for most classic target shooters. But I'm not alone in finding that .45-70 tends to beat up a person if more than 10 to 15 rounds are going to be shot in a day. On the other hand .38-55 is softer enough that I can shoot a whole box of 100 if needed and still smile at the end of the day. So while it's more work to get to where you need to be to get the best out of your old rifle my opinion is that it's well worth the work and cost to get set up for reloading even if it's only on a small scale.