One part done: lowering the bar numbers:

MWE
\documentclass[tikz,border=3mm]{standalone}
\usepackage[top=3cm, bottom=2.8cm, left=2.5cm, right=0.5cm, marginparwidth=0cm, marginparsep=0cm, headheight=13.6pt, a4paper]{geometry}
\usepackage[T1]{fontenc}
\usepackage{polyglossia}
\setmainlanguage{spanish}
\usepackage{xcolor}
\usepackage{musixtex}
\begin{document}
\vspace{5cm}
\begin{music}
\makeatletter
\def\writethebarno{\raisebox{-16\internote}{{\color{red}\fontbarno\large\the\barno\kernm\qn@width}}}
\makeatother
\font\A=phvb8t at 8pt% added for letter type.
\font\B=phvb8t at 8pt% added for letter type.
\setclef1\treble % Clave de Sol para el Instrumento 1
\nostartrule % Inicia sin barra vertical
\generalmeter{\meterfrac{4}4}
\hbox to\hsize\bgroup\vbox\bgroup\hsize=4.2in % specify length of left segment
\startpiece
\def\fontbarno{\B\bf\scriptsize}
\Notes\hu{gg}\en\bar
\znotes\zcn{s}{\hspace{0.12cm}\bf\A To Coda}\en
\Notes\qu{gg}\hu{g}\en\bar
\znotes\hspace{-0.25cm}\coda q\en
\notes\qu{g}\Dqbu gg\qu{gg}\en\bar
\znotes\ccn{s}{\hspace{+3.7cm}\bf\A D.C. al Coda}\en
\NOtes\qu{g}\qp\hp\en
\setdoublebar
\endpiece% Se finaliza el fragmento musical
\egroup\hfill\vbox\bgroup\hsize=2.2in
\startbarno=5
\startpiece
\def\fontbarno{\B\bf\scriptsize}
\znotes\coda o\en
\notes\qu{gg}\hu{g}\en\bar
\notes\hu{gg}\en\bar
\Notes\wh{g}\en\setdoubleBAR\endpiece
\egroup\egroup% close open \bgroups
\end{music}
\end{document}
Edit
Raw first bar number:

MWE
\documentclass[tikz,border=3mm]{standalone}
\usepackage[top=3cm, bottom=2.8cm, left=2.5cm, right=0.5cm, marginparwidth=0cm, marginparsep=0cm, headheight=13.6pt, a4paper]{geometry}
\usepackage[T1]{fontenc}
\usepackage{polyglossia}
\setmainlanguage{spanish}
\usepackage{xcolor}
\usepackage{musixtex}
\makeatletter
\def\piece@beginb{\advance\line@width-\parindent \s@indent\parindent
\writ@staffs \writ@names \wbarno@x\advance\line@width\parindent}
\def\writethebarno{\raisebox{-16\internote}{{\color{red}\fontbarno\large\the\barno\kernm\qn@width}}}
\makeatother
\begin{document}
\vspace{5cm}
\begin{music}
\font\A=phvb8t at 8pt% added for letter type.
\font\B=phvb8t at 8pt% added for letter type.
\setclef1\treble % Clave de Sol para el Instrumento 1
\nostartrule % Inicia sin barra vertical
\generalmeter{\meterfrac{4}4}
\hbox to\hsize\bgroup\vbox\bgroup\hsize=4.2in % specify length of left segment
\startpiece
\def\fontbarno{\B\bf\scriptsize}
\Notes\hu{gg}\en\bar
\znotes\zcn{s}{\hspace{0.12cm}\bf\A To Coda}\en
\Notes\qu{gg}\hu{g}\en\bar
\znotes\hspace{-0.25cm}\coda q\en
\notes\qu{g}\Dqbu gg\qu{gg}\en\bar
\znotes\ccn{s}{\hspace{+3.7cm}\bf\A D.C. al Coda}\en
\Notes\qu{g}\qp\hp\en
\setdoublebar
\endpiece% Se finaliza el fragmento musical
\egroup\hfill\vbox\bgroup\hsize=2.2in
\startbarno=5
\startpiece
\def\fontbarno{\B\bf\scriptsize}
\znotes\coda o\en
\notes\qu{gg}\hu{g}\en\bar
\notes\hu{gg}\en\bar
\Notes\wh{g}\en\setdoubleBAR\endpiece
\egroup\egroup% close open \bgroups
\end{music}
\end{document}
Edit 2
Moving the first bar number right:

A total hack. See comments in the code.
MWE
\documentclass[tikz,border=3mm]{standalone}
\usepackage[top=3cm, bottom=2.8cm, left=2.5cm, right=0.5cm, marginparwidth=0cm, marginparsep=0cm, headheight=13.6pt, a4paper]{geometry}
\usepackage[T1]{fontenc}
\usepackage{polyglossia}
\setmainlanguage{spanish}
\usepackage{xcolor}
\usepackage{musixtex}
%*** Code copied from musixtex.tex
%*** redefine \fontbarno
\def\fontbarno{\color{red}\large\bfseries}
\makeatletter
%*** for the first bar number
%*** clone of \wbarno@x
%*** points to \writethebarnob
\def\wbarno@xb{%
\ifx\volta@set\empty
\n@iii\barno \divide\n@iii\freqbarno\relax
\ifnum\n@iii=\lastbarno
\else \y@\altplancher \advance\y@\systemheight \advance\y@\tw@\internote
\raise\y@\llap{\writethebarnob}\fi
\else
\ifx\volta@cut\empty
\else % volta was cut, there is nothing colliding with bar number.
\n@iii\barno \divide\n@iii\freqbarno\relax
\ifnum\n@iii=\lastbarno
\else \y@\altplancher \advance\y@\systemheight \advance\y@\tw@\internote
\raise\y@\llap{\writethebarno}\fi
\fi
\write@volta
\fi
\ifx\pdl@cut\empty % 1.21 RDT
\else% pedal rule was cut at the end of the previous line
\global\pdl@pos\lin@pos\let\pdl@cut\empty % reset \ped@pos
\fi
\b@rrul@ %+simick
\n@iii\barno \divide\n@iii\freqbarno\relax \lastbarno\n@iii }
%*** redefine \piece@beginb
%*** points to \wbarno@xb
\def\piece@beginb{\advance\line@width-\parindent \s@indent\parindent
\writ@staffs \writ@names \wbarno@xb\advance\line@width\parindent}
%*** redefine \writethebarno
%*** lower the number with \raisebox
%*** make the number \large and red
\def\writethebarno{\raisebox{-16\internote}{{\color{red}\fontbarno\large\the\barno\kernm\qn@width}}}
%*** clone of \writethebarno
%*** for moving (first) bar number right, -3em
\def\writethebarnob{\raisebox{-16\internote}{{\color{red}\fontbarno\large\the\barno\kern-3em}}}
\makeatother
\begin{document}
\vspace{5cm}
\begin{music}
\font\A=phvb8t at 8pt% added for letter type.
\font\B=phvb8t at 8pt% added for letter type.
\setclef1\treble % Clave de Sol para el Instrumento 1
\nostartrule % Inicia sin barra vertical
\generalmeter{\meterfrac{4}4}
\hbox to\hsize\bgroup\vbox\bgroup\hsize=4.2in % specify length of left segment
\startpiece
\def\fontbarno{\B\bf\scriptsize}
\Notes\hu{gg}\en\bar
\znotes\zcn{s}{\hspace{0.12cm}\bf\A To Coda}\en
\Notes\qu{gg}\hu{g}\en\bar
\znotes\hspace{-0.25cm}\coda q\en
\notes\qu{g}\Dqbu gg\qu{gg}\en\bar
\znotes\ccn{s}{\hspace{+3.7cm}\bf\A D.C. al Coda}\en
\Notes\qu{g}\qp\hp\en
\setdoublebar
\endpiece% Se finaliza el fragmento musical
\egroup\hfill\vbox\bgroup\hsize=2.2in
\startbarno=5
\startpiece
\def\fontbarno{\B\bf\scriptsize}
\znotes\coda o\en
\notes\qu{gg}\hu{g}\en\bar
\notes\hu{gg}\en\bar
\Notes\wh{g}\en\setdoubleBAR\endpiece
\egroup\egroup% close open \bgroups
\end{music}
\end{document}
As remarked in the comments, this is not an enhancement feature-add to the whole MusiXTeX program, but a one-off mod to one part of one strand of it.