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There are several issues with your test document:
Most importantly, place the \bibliography directive not in the preamble, but inside the document environment, where you want the formatted bibliography to be typeset. Usually, this is toward the end of the document.
The jcappub package loads natbib automatically. If you want authoryear-style citation call-...
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There is no natbib option to do this. You can only influence the alphabetic label produced by alpha if you modify (a copy of) alpha.bst directly.
(Note that alpha isn't even a natbib style and that natbib supports mainly numeric and author-year citations, but not alphabetic styles. It probably works well enough if you don't try to use \citet or \citeauthor ...
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In order to get the numeric-style citation call-outs sorted, you need to load the natbib citation management package with the option sort&compress:
\usepackage[sort&compress,<any other options>]{natbib}
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Here might be a solution for those of you receiving similar warnings and errors even though you've uploaded a correct bbl file (as comments to the accepted answer suggests).
I got similar errors and warnings, but I've uploaded the correct bbl file as requested by arXiv. However, I was getting a lot of warning messages for references and they all were ...
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