All of the above, basically. If it's a protein antigen, you can perform an immunoprecipitation with the auto-antibody and use any one (or more) of several methods (Western blot, various methods of sequencing, etc.) to identify the binding partner(s). DNA and RNA can be identified easily by biochemical means. You can also use ELISAs or similar methods like electrochemiluminescence which allow a greater degree of multiplexing.
Sequencing the autoantibodies' variable regions may give some information about the structure of the antigen, but not enough to actually identify the binding partner. Also, one key fact to keep in mind is that the antibody response in many autoimmune diseases is polyclonal (just as the T cell response often is), meaning there could be dozens to hundreds of different complementarity-determining regions (CDRs) to sequence, a very laborious process that may not yield much information.
Immunohistochemistry (IHC) can be useful for seeing which cell types the antibodies bind to, but it doesn't give great resolution on the subcellular level, and doesn't tell you anything about what the actual target is. Additionally, due to the amount of chemical processing a formalin-fixed paraffin-embedded (FFPE) tissue goes through, the antibody's antigen may not be exposed properly for the antibody to bind.
From what I can find, people suspected an association between ANCA and lupus due to the similarities in both diseases' clinical presentation. Here is a paper from 1996 talking about it, there are many others.