There is still a lot of mystery surrounding artistic practice in ancient Rome. Your training would depend on what kind of artist you are hoping to be, and therefore what kind of workshop you will join: will you sculpt/carve lamp molds, paint wall decorations or panels, carve stone, lay mosaics, etc.? I'm going to go with paintings, since it sounds like you're interested in sketching images. The short answer is that you're probably not going to practice on papyrus.
Who were artists?: We hear of a few named famous artists--mostly sculptors, architects, and panel painters, and mostly Greeks whose works elite Romans just loved--and we have a few instances of partially finished artworks (like wall paintings) that shed some light on the potential processes of creating art. There is also a strong tradition of connoisseurship that scholars use to classify and track particular artists or workshops whose names are unknown (in the tradition of Sir. John Beazley's classification of Athenian vase painters by identifying the "hands", basically stylistic quirks, of different artists who did not sign their work and assigning them names like "The Berlin Painter").
Pliny the Elder's Natural Histories devotes a lot of space to famous artists, grouped by medium (book 35, chapter 36 is all about painters), and at some point touches on artists being trained by other successful artists but is often vague about it. Zeuxis, for example, a 5th century BCE painter from southern Italy famous for his ability to deceive birds into trying to eat the grapes he painted, was trained by either Demophilus of Himera or Neseus of Thasos (neither of whom garner any other mention in the work).
There is a really interesting passage a bit later that refers to the development of a systematized artistic education in ancient Greece during the fourth century BCE. He mentions first the establishment of three artistic schools--the Ionian, Sicyonian, and Attic--and then introduces Pamphilus of Amphipolis, a painter who emphasized mathematical and scientific skills in painting and charged hefty fees to his pupils (including future very famous painter Apelles). Pliny notes that "It was through his influence that, first at Sicyon, and then throughout the whole of Greece, all children of free birth were taught the graphic art, or in other words, the art of depicting on boxwood, before all others" (Pliny HN 35.36). So, in the 300's BCE, you would find Greek youths practicing their drawing on wooden boards (not papyrus).
The thing about Pliny that makes him a slightly less useful source for answering your question is that he gets even vaguer after the period of Alexander the Great. For Roman painters, he devotes just a fraction of the space he does to the Greek masters, and mentions nothing about their training. Pliny was a little bit critical of the Roman fashion for painting houses (for the eyes of the few) instead of moveable panels, and seems to imply that Romans didn't value art as a skill quite as much. By the Roman period, it seems that there were fewer famous artists and more of a large-scale workshop model. Less artist, more artisan.
How would a wall painting workshop work? Often in Roman wall paintings, you can see seams called "giornate" that seem to indicate how much of a wall artists could paint in fresco during a day. (Roman wall paintings seem to be a mix of fresco and secco methods, for the most part). Some scholars think that artists in training would tackle the simpler elements, borders and framing devices, while the lead artist(s) of the workshop would work on more complex parts, like central scenes and figures. There are a lot of formulaic elements that show up again and again (and change over time). In the Augustan period, around year 1 as in your question, the "Third Pompeian Style" was all the rage, with large surfaces of flat color, delicate framing elements often in the from of attenuated architecture, and often with large central panel pictures in the middle of each wall showing a mythological or landscape scene. The replication of some elements even within these more "painterly" scenes in the middle of the wall indicates that copybooks were likely in circulation that artists could use to base their work around. But as far as the details of workshop organization go--whether they moved around or were local, family based, hired, etc.--these are still hazy.
In general, in Roman aesthetics, riffing off of the work of a famous innovative artist was popular. One of the few Roman painters Pliny mentions in detail is Ludius (of the Augustan period) who was the first to paint walls with scenes of maritime cityscapes and populated landscapes. Both of these subjects were picked up by other artists and appear in many houses around the Bay of Naples, replicated, refashioned, and recontextualized in new places by new artists who did not sign their work. The same went for sculptures--artisans recreating certain types popularized by Greek artists like Polykleitos' Doryphoros (Spear bearer)--and mosaics--as in the replication of Sosus' famous "Unswept Floor" scheme and his image of doves on the lip of a bowl across multiple contexts.
All of this to say that, as an average workshop artisan, you're probably going to be training by first helping with preparation, learning simple and repetitive motifs, and then copying the works of popular artists, then finally putting your own slight spin on familiar subjects. Eventually, if you made a name for yourself, you might get a commission that would let you be extra innovative, maybe.
What surfaces were available for practicing art?: Now, let's say you just want to draw for your own creative expression outside the strictures of your workshop, so you have to find your own materials. We have drawings preserved on just about everything from the Roman period: graffiti scratched into or drawn on walls (a particular favorite of mine is this hairy moth from Oplontis drawn in charcoal beneath a stairwell to cramped probable slave quarters), ostraka (broken bits of pottery reused for writing or doodling surfaces), even a reusable wax tablet or charcoal on a piece of wood or stone. We do have a few doodles on papyrus (it's much later, but this elephant is very cute) and evidence of painted scrolls (from triumphal processions, and some see the reliefs on the Column of Trajan as a representation of an illustrated scroll in stone). So there are a lot of options for you, and even if you have no money or status and aren't actually an artist by trade, you still might take some time to draw a boat on a kitchen wall for fun.
As for the price of papyrus, my information might be out of date (Skeat 1995, available on JStor) but it seems papyrus might be less expensive than often thought, based on being sold in larger quantities than often thought (a roll vs. a sheet). Nonetheless, the expense wasn't trivial, and papyrus reuse is attested often in the form of palimpsests and writing on both sides. But the fact that we have so many papyri from dumps, like at Oxyrhynchus in Egypt, that show it being used for a large variety of things like personal correspondence, record keeping, wills, etc., makes it seem that papyrus was widely used, but for permanent/semi-permanent purposes. Not so much for things that didn't need to be kept, at least in the medium-term.
(Edited because I hit send before I was done)