Citing author or year only without natbib

Whether using biblatex is allowed in the solution or not is unclear. However, if biblatex is indeed allowed, I recommend using biblatex's \textcite (as well as \citeauthor and \citeyear), with the following tweaks:

\documentclass{article}

\usepackage{filecontents}
\begin{filecontents*}{mybiblio.bib}
@Book{Batchelor:2000,
    address     = {Cambridge, UK},
    author      = {Batchelor, G.{\,}K.},
    title       = {An Introduction to Fluid Dynamics},
    publisher   = {Cambridge University Press},
    year        = {2000},
}
\end{filecontents*}

\usepackage[style=authoryear]{biblatex}
\renewcommand*\bibopenparen{[}
\renewcommand*\bibcloseparen{]}
\addbibresource{mybiblio}

\begin{document}
    \noindent
    All of this is discussed in \textcite{Batchelor:2000}.\\
    \citeauthor{Batchelor:2000}'s book was published in \citeyear{Batchelor:2000}.
    \printbibliography
\end{document}

Use biblatex, and the command \citeyear.