Insert a period / full stop if caption argument doesn't end with one

It's as easy as loading amsthm, which defines an \@addpunct command, which will insert its argument only if no punctuation precedes it.

\documentclass{scrartcl}
\usepackage{amsthm}% yes, that's it!
\usepackage{caption}
\usepackage{graphicx}

\makeatletter
\newcommand{\Notes}[1]{\captionsetup{position=below}\caption*{#1\@addpunct{.}}}
\makeatother

\begin{document}

\begin{figure}[htp]
\centering
\caption{Actual caption}
\includegraphics[width=4cm]{example-image}
\Notes{Notes}
\end{figure}

\begin{figure}[htp]
\centering
\caption{Another caption}
\includegraphics[width=4cm]{example-image}
\Notes{Notes with full stop.}
\end{figure}

\begin{figure}[htp]
\centering
\caption{Another caption}
\includegraphics[width=4cm]{example-image}
\Notes{Notes with exclamation mark!}
\end{figure}

\end{document}

The following example evaluates the space factor at the end of the caption to detect, if there was a punctuation char before.

\documentclass[a5paper]{scrartcl}
\usepackage[paperheight=7cm]{geometry}% for smaller image in answer for TeX.SX
\usepackage{caption}
\usepackage{graphicx}

\newcommand{\Notes}[1]{\captionsetup{position=below}\caption*{#1}}

\makeatletter
\newcommand*{\RedefCaption}{%
  \let\org@caption\@caption
  \let\@caption\dot@caption
}
\def\dot@caption#1[#2]#3{%
  \dot@test@def\tmp@a{#2}%
  \dot@test@def\tmp@b{#3}%
  \org@caption{#1}[\tmp@a]\tmp@b
}
\newcommand*{\dot@test@def}[2]{%
  \begingroup
    % Set space factor to values larger than 1000 for punctuation characters
    \nonfrenchspacing
    % Set space factor to 1000 for upper case letter, which usually
    % have 999. This supports abbreviations: After an uppercase letter
    % the space factor is 999, a period would want it to set to 3000,
    % but TeX does not increase over 1000 in one step. Thus, the
    % space factor is 1000 after "A.".
    % Here, we do not want to support abbreviations, because
    % a period needs to be omitted, if there was a dot before,
    % regardless if the previous dot marks the end of the sentence
    % or belongs to an abbreviation.
    \count@=`\@\relax
    \@whilenum\count@<`\Z\do{%
      \advance\count@\@ne % \@ne = 1
      \sfcode\count@=\@m % \@m = 1000
    }%
    \sbox0{#2%
      \ifnum\spacefactor>\@m
        \global\let\dot@found=Y%
      \else
        \global\let\dot@found=N%
      \fi
    }%
  \endgroup
  \if\dot@found Y%
    \def#1{#2}%
  \else
    \def#1{#2.}%
  \fi
}
\AtBeginDocument{\RedefCaption}
\makeatother

\begin{document}

\begin{figure}
\caption{Actual caption}
% \includegraphics{Example-image}
\Notes{Notes}
\end{figure}

\begin{figure}
\caption{Another caption}
% \includegraphics{Example-image}
\Notes{Notes with full stop.}
\end{figure}

\end{document}

(Your requirement wasn't entirely clear to me. I assume that you want periods (full stops) at the ends of the arguments of both \caption and \Notes. If it's only needed for \Notes, I trust you'll be able to figure out how to modify the code shown below.)

Here's a LuaLaTeX-based solution. It adds missing periods ("full stops") at the end of the arguments of \caption and \Notes instructions. It works by scanning the input, at a very early stage of processing, for instances of \caption{...} and \Notes{...} and examining the arguments; if they lack a period at the end, a period is inserted.

The only requirement for the input stream is that the commands \caption and \Notes and their respective arguments all be on the same line. I trust that this doesn't constitute a binding restriction.

%% Compile with LuaLaTeX
\documentclass{scrartcl}
\usepackage{caption}

\usepackage{luacode}
\begin{luacode*}
function add_period ( s )
  if string.find ( s , "\\caption%s-{" ) then
    s = string.gsub ( s , "(\\caption)%s-(%b{})", 
           function ( capt , argu )
              argu = string.sub ( argu , 2 , -2 )
              if not string.find ( argu , "%.%s-$" ) then 
                argu = argu .. "."
              end
              return ( capt .. "{" .. argu .. "}" )
           end )
  end
  if string.find ( s , "\\Notes%s-{" ) then
    s = string.gsub ( s , "(\\Notes)%s-(%b{})", 
           function ( capt , argu )
              argu = string.sub ( argu , 2 , -2 )
              if not string.find ( argu , "%.%s-$" ) then 
                argu = argu .. "."
              end
              return ( capt .. "{" .. argu .. "}" )
           end )
  end
  return ( s ) 
end
luatexbase.add_to_callback( "process_input_buffer" , add_period , "add_period" )
\end{luacode*}        

\newcommand{\Notes}[1]{\captionsetup{position=below}\caption*{#1}}

\begin{document}

\begin{figure}[ht!]
\caption {Actual caption}
\Notes{Notes}
\end{figure}

\begin{figure}[h!]
\caption {Another caption. } \Notes {Notes with full stop. }
\end{figure}

\end{document}