KOMA-Script scrbook: How to change V-space between chapter prefix and title

The scrbook class introduces an additional \vskip of .5\baselineskip between the heading "Chapter" and the title; unfortunately this skip is hard-coded, so you will have to redefine \@@makechapterhead to suppress it:

\documentclass[12pt,letterpaper,oneside,chapterprefix=on]{scrbook}

%Margins
\usepackage[letterpaper,left=1.5in,top=1in,right=1in,bottom=1in]{geometry}

%Type
\usepackage{fontspec}
%\setmainfont{Times New Roman}
\usepackage{unicode-math}
\setmathfont{xits-math.otf}
\usepackage[doublespacing]{setspace}
\setlength\parindent{0.5in}

\usepackage{microtype}
\usepackage{lipsum}

%Remove vertical space above/below chapter headings:
%http://tex.stackexchange.com/questions/43087/remove-space-before-chapter-title-with-koma-      script-scrbook
\renewcommand*{\chapterheadstartvskip}{\vspace*{-\baselineskip}}
\renewcommand*{\chapterheadendvskip}{\vspace*{0in}}
\renewcommand*{\chapterformat}{\chapappifchapterprefix{\nobreakspace}\thechapter}

\setkomafont{disposition}{\normalcolor\rmfamily}
\setkomafont{chapter}{\centering\MakeUppercase}

\makeatletter
\renewcommand*{\@@makechapterhead}[1]{\chapterheadstartvskip
  {%
    \setlength{\parindent}{\z@}\setlength{\parfillskip}{\fill}%
    \normalfont\sectfont\nobreak\size@chapter{}%
    \if@chapterprefix
      \let\@tempa\raggedsection
    \else
      \let\@tempa\@hangfrom
    \fi
    \@tempa{\ifnum \c@secnumdepth >\m@ne%
        \if@mainmatter
          \if@chapterprefix
            \expandafter\size@chapterprefix
          \else
            \expandafter\size@chapter
          \fi
          {\chapterformat}%
          \if@chapterprefix
            \size@chapterprefix{}\endgraf\nobreak%\vskip\baselineskip
          \fi
        \fi
      \fi
    }%
    {\raggedsection \interlinepenalty\@M\size@chapter{#1}\par}}%
  \nobreak\chapterheadendvskip
}
\makeatother

\begin{document}

\frontmatter

\chapter*{Abstract Title Page}
\chapter*{Abstract}
\chapter*{Copright Page}
\chapter*{Title Page}

\tableofcontents
\listoffigures

\mainmatter

\chapter{Introduction}
\lipsum[1-2]
\appendix
\backmatter

\end{document}

With the etoolbox package, one can do this in one-line patching \@@makechapterhead:

\documentclass[12pt,letterpaper,oneside,chapterprefix=on]{scrbook}

%Margins
\usepackage[letterpaper,left=1.5in,top=1in,right=1in,bottom=1in]{geometry}

%Type
\usepackage{fontspec}
%\setmainfont{Times New Roman}
\usepackage{unicode-math}
\setmathfont{xits-math.otf}
\usepackage[doublespacing]{setspace}
\setlength\parindent{0.5in}

\usepackage{microtype}
\usepackage{etoolbox}
\usepackage{lipsum}

%Remove vertical space above/below chapter headings:
%http://tex.stackexchange.com/questions/43087/remove-space-before-chapter-title-with-koma-      script-scrbook
\renewcommand*{\chapterheadstartvskip}{\vspace*{-\baselineskip}}
\renewcommand*{\chapterheadendvskip}{\vspace*{0in}}
\renewcommand*{\chapterformat}{\chapappifchapterprefix{\nobreakspace}\thechapter}

\setkomafont{disposition}{\normalcolor\rmfamily}
\setkomafont{chapter}{\centering\MakeUppercase}

\makeatletter
\patchcmd{\@@makechapterhead}{\vskip.5\baselineskip}{}{}{}
\makeatother

\begin{document}

\frontmatter

\chapter*{Abstract Title Page}
\chapter*{Abstract}
\chapter*{Copright Page}
\chapter*{Title Page}

\tableofcontents
\listoffigures

\mainmatter

\chapter{Introduction}
\lipsum[1-2]
\appendix
\backmatter

\end{document}

I commented out the line \setmainfont{Times New Roman} since I don't have that font in my system.

As egreg noticed in a comment, with the settings of the question, the heading "Chapter #" is slightly shifted to the left; to correct this, the following additional redefinition had to be added:

\renewcommand*{\chapterformat}{\chapappifchapterprefix{\nobreakspace}\thechapter}

With a KOMA-Script class use \RedeclareSectionCommand[...]{chapter} to change the skips before, after and inside the chapter heading:

\RedeclareSectionCommand[
  afterindent=false,beforeskip=0pt,% afterindent needs version 3.26 or newer
  afterskip=0pt,innerskip=0pt
]{chapter}

or

\RedeclareSectionCommand[
  beforeskip=-1sp,
  afterskip=0pt,innerskip=0pt
]{chapter}

Do not use \MakeUppercase as a font command. Its usage in the second argument of \setkomafont or \addtokomafont results in an error.

You can redefine \chapterlineswithprefixformat (used for chapter headings with chapterprefix=on) and \chapterlinesformat (used for chapter headings with chapterprefix=off, default) to get chapter headings in capital letters:

\renewcommand\chapterlineswithprefixformat[3]{%
  \ifstr{#1}{chapter}
    {\MakeUppercase{#2#3}}
    {#2#3}%
}
\makeatletter
\renewcommand\chapterlinesformat[3]{%
  \ifstr{#1}{chapter}
    {\@hangfrom{#2}{\MakeUppercase{#3}}}
    {\@hangfrom{#2}{#3}}%
}
\makeatother

Example

\documentclass[12pt,letterpaper,oneside,chapterprefix=on]{scrbook}

%Margins
\usepackage[letterpaper,left=1.5in,top=1in,right=1in,bottom=1in]{geometry}

%Type
\usepackage{fontspec}
\setmainfont{Times New Roman}
\usepackage{unicode-math}
\setmathfont{xits-math.otf}
\usepackage[doublespacing]{setspace}
\setlength\parindent{0.5in}

\usepackage{microtype}
\usepackage{lipsum}

\RedeclareSectionCommand[
  afterindent=false,beforeskip=0pt,% afterindent needs version 3.26 or newer
  afterskip=0pt,innerskip=0pt
]{chapter}

\setkomafont{disposition}{\normalcolor\rmfamily}
\setkomafont{chapter}{}
\renewcommand*\raggedchapter{\centering}

\renewcommand\chapterlineswithprefixformat[3]{%
  \ifstr{#1}{chapter}
    {\MakeUppercase{#2#3}}
    {#2#3}%
}

%\makeatletter
%\renewcommand\chapterlinesformat[3]{%
  %\ifstr{#1}{chapter}
    %{\@hangfrom{#2}{\MakeUppercase{#3}}}
    %{\@hangfrom{#2}{#3}}%
%}
%\makeatother

\begin{document}
\frontmatter
\addchap*{Abstract Title Page}
\addchap*{Abstract}
\addchap*{Copright Page}
\addchap*{Title Page}
\tableofcontents
\listoffigures
\mainmatter
\chapter{Introduction}
\lipsum[1-2]
\chapter{Chapter 2}
\lipsum[1-2]
\appendix
\backmatter
\end{document}