How to make last names of authors appear first in my bibliography?

You need to change a single line in the bibliography style file to achieve your formatting objective. I suggest you proceed as follows:

• Locate the file plainnat.bst in your TeX distribution. Make a copy of this file, and call the copy, say, plainnat-reversed.bst. (Do not edit an original file of the TeX distribution directly.

• Open the file plainnat-reversed.bst in a text editor. The text editor program you use to edit your tex files will do fine.

• In the file plainnat-reversed.bst, locate the function format.names. (It starts on line 216 in my copy of the file.)

• Inside this function, locate the following line:

      { s nameptr "{ff~}{vv~}{ll}{, jj}" format.name$ 't :=
  

  Change this line to

      { s nameptr "{vv~}{ll}{, jj}{, ff}" format.name$ 't :=
  

  If you also want to truncate the given names to just their initials, change the line to

      { s nameptr "{vv~}{ll}{, jj}{, f.}" format.name$ 't :=
  

• Save the file plainnat-reversed.bst either in the directory where you main tex file is located or in a directory that's searched by BibTeX. If you choose the second option, be sure to update the filename database of your TeX distribution suitably.

• In your main tex file, change the instruction \bibliographystyle{plainnat} to \bibliographystyle{plainnat-reversed}. Rerun LaTeX, BibTeX, and LaTeX twice more to fully propagate all changes.

Happy BibTeXing!

A full MWE:

\RequirePackage{filecontents}
\begin{filecontents}{mybib.bib}
@article{battin2009boundless,
  title={The boundless carbon cycle},
  author={Battin, Tom J. and Luyssaert, Sebastiaan and Kaplan, Louis A. and Aufdenkampe, Anthony K. and Richter, Andreas and Tranvik, Lars J.},
  journal={Nature Geoscience},
  volume={2},
  number={9},
  pages={598--600},
  year={2009},
  publisher={Nature Publishing Group}
}
\end{filecontents}

\documentclass[12pt]{article}
\usepackage[round]{natbib}
\bibliographystyle{plainnat-reversed}

\begin{document}
\cite{battin2009boundless}
\bibliography{mybib}   
\end{document}

based on https://tex.stackexchange.com/a/112341/36296 and comments from moewe

You can mimic a similar style in biblatex

\documentclass{article}

\usepackage{filecontents} %only for this example
\begin{filecontents}{\jobname.bib}
@article{battin2009boundless,
  title={The boundless carbon cycle},
  author={Battin, Tom J and Luyssaert, Sebastiaan and Kaplan, Louis A and Aufdenkampe, Anthony K and Richter, Andreas and Tranvik, Lars J},
  journal={Nature Geoscience},
  volume={2},
  number={9},
  pages={598--600},
  year={2009},
  publisher={Nature Publishing Group}
}
\end{filecontents}

\usepackage[%
    style=authoryear,
    giveninits=true,
    natbib=true,
    maxbibnames=99,
    uniquename=init
]{biblatex}

\DeclareNameAlias{sortname}{family-given}
\renewcommand*{\multinamedelim}{\addcomma\space}
\renewcommand*{\finalnamedelim}{\addcomma\space}
\renewcommand*{\nameyeardelim}{\addcomma\space}
\setlength{\bibitemsep}{\baselineskip}
\renewbibmacro{in:}{}

\addbibresource{\jobname.bib}

\begin{document}
\nocite{*}
\printbibliography
\end{document}

(in case the exact formatting of page numbers etc. is important to you, they can be done exactly as in your example, just a bit of extra work)