To get the desired ordering of authors' first and last names, I suggest you do the following:
Find the file abbrvnat.bst
in your TeX distribution. Make a copy of this file and name the copy, say, myabbrvnat.bst
. (Don't edit a file from the TeX distribution directly.
Open the file myabbrvnat.bst
in a text editor; the editor you use to edit your tex files will do fine.
Find the function format.names
in the file. (In my copy of the file, this function starts on line 216. Delete all 27 or so lines of function; the last line will contain just a right curly brace, }
.
Insert the following code chunk in the place of the deleted function:
FUNCTION {format.names}
{ duplicate$ empty$ 'skip$ {
's :=
"" 't :=
#1 'nameptr :=
s num.names$ 'numnames :=
numnames 'namesleft :=
{ namesleft #0 > }
{ s nameptr
duplicate$ #1 >
{ "{f.~}{vv~}{ll}{, jj}" }
{ "{vv~}{ll}{, f.}{, jj}" }
if$
format.name$
't :=
nameptr #1 >
{
namesleft #1 >
{ ", " * t * }
{
s nameptr "{ll}" format.name$ duplicate$ "others" =
{ 't := }
{ pop$ }
if$
"," *
t "others" =
{ " et~al." * }
{ " " * t * }
if$
}
if$
}
't
if$
nameptr #1 + 'nameptr :=
namesleft #1 - 'namesleft :=
}
while$
} if$
}
Do note that I've interpreted your specification as indicating that there should be no "and" connector word before the final author in the author list.
Save the file myabbrvnat.bst
, either in the directory where your main tex file is located or in a folder that's searched by BibTeX. If you choose the latter method, be sure to udpate the filename database of your TeX distribution appropriately.
Start using the new bibliography style by replacing the instruction \bibliographystyle{abbrvnat}
with \bibliographystyle{myabbrvnat}
. Be sure to run LaTeX, BibTeX, and LaTeX twice more to fully propagate all changes.
Here's an example file:

\documentclass{article}
\usepackage[numbers]{natbib}
\bibliographystyle{myabbrvnat}
\usepackage{filecontents}
\begin{filecontents*}{mybib.bib}
@article{abc,
author = "Anna Anderson and Brenda Branson and Carla Carlson",
title = "Random thoughts",
journal = "Circularity Today",
year = 3001,
volume = 1,
number = 1,
pages = "1-100",
}
\end{filecontents*}
\begin{document}
\nocite{*}
\bibliography{mybib}
\end{document}