Giving a simple answer is difficult as the field of humoral immunology is quite extensive. Here is an abbreviated overview.
Our body produces myriads of different antibodies during an immune response, not just one. It is impractical to isolate and sequence one antibody protein as their sequence differs in very subtle ways. N-terminal sequencing like Edman degradation is not practical as it covers only a few dozen residues at most. Perhaps proteolytic digest of an antibody and peptide sequencing is possible, but again this requires a large amount of a pure antibody. Protein sequencing is not practical.
Instead, people sequence the DNA corresponding to a given antibody instead of sequencing the antibody protein itself. As a given B cell expresses one antibody in large amounts, Milstein and Kohler (Nobel laureates) devised a technique to fuse B cells with tumor cells, creating hybridomas. Each hybridoma thus produces a single antibody. Such hybridomas may be cultured – monocultured in fact - in the lab to large amounts, leading to the production of a uniform, single antibody, termed monoclonal antibody. The corresponding DNA sequence of such monoclonal antibody may be readily sequenced.
I would think it feasible to create a library of individual hybridoma monocultures using a patient’s B cells, and then select the hybridoma producing the best monoclonal antibody from the library, and sequence that DNA. Epitope binning [https://en.wikipedia.org/wiki/Epitope_binning] is one example of a technique to identify the best antibody. The successful hybridoma may then be culture to large scale to produce lots of that exact antibody.
Administering such purified antibodies to a patient is not straightforward as this foreign protein will be itself recognized by the patient's immune system and cleared. There are techniques to reduce the immunogenicity of a given antibody, but that will be a lengthy discussion in itself.
