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}
natbibcitation management package, the stylesabbrvnatandmyabbrvnatwill automatically truncate the citation callout to FirstAuthor et al for entries with three or more authors. – Mico Sep 04 '14 at 11:24unsrtstyle, but the instructions apply to other styles as well, includingabbrvandabbrvnat. – Mico Sep 04 '14 at 15:49bibinfo.check, not sure if the currentabbrevnat.bstis newer than the version this question is made for. Any suggestions? – daleif Feb 27 '17 at 13:33makebstutility.) Let me get back to you later today with a full answer. In the meantime, simply deleting the lines (strings)'bibinfo :=andbibinfo bibinfo.checkshould get you going -- assuming all author names are well-formed. I suspect thebibinfo.checkfunction is there to issue an alert if one or more names are not well formed. I need to figure out what a suitable version of the functionbibinfo.checkshould look like. – Mico Feb 27 '17 at 14:41bibinfoandbibinfo.checkin the replacement code. – Mico Feb 27 '17 at 18:34