Like the vast majority of ancient writings that survive, it was copied - many times. Julia Haig Gaisser lists 22 manuscripts written before the 15th century. Most of these date from the 14th century; the oldest seems to be from the late 11th. Most of these were produced by monks, in scriptoria (dedicated workshops for the copying and production of manuscripts). A few, like the British Library's Harley 4838, were probably made by private bookmakers who ran workshops or ateliers of their own and produced books for profit. Per Gaisser, all of these copies stem ultimately from one single copy made in 395-7 by the pagan writer Sallustius. It probably spend the next few hundred years in a monastic library somewhere between Rome and Naples before arriving at the (also monastic) library of Monte Cassino, where it was copied in the 11th century (that old manuscript I mentioned above). All of the other manuscripts that we have are either copies of that copy, or copies of copies of that copy, and so on.

The original original was probably written on papyrus. Sallustius' copy was almost certainly on parchment, which by the 4th century had mostly replaced papyrus as the standard writing material.

Mot of this comes from Gaisser's The fortunes of Apuleius and the Golden Ass: a study in transmission and reception, especially pp. 60-65. Appendix 2 lists all extant manuscripts. Appendix 3 lists all MSS from before 1400.