You can use the current page family of nodes to avoid manually specifying coordinates:
\documentclass[a4paper,12pt,twoside]{article}
\usepackage[norsk]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[margin=2.5cm]{geometry}
\usepackage{setspace}
\onehalfspacing
\usepackage{parskip}
\usepackage{fontawesome}
\usepackage{graphicx,tikz}
\definecolor{color1}{rgb}{0.22,0.45,0.70} %blå fra cvmodern.
\usepackage{lipsum}
\begin{document}
\setcounter{page}{0}
\pagenumbering{roman}
%Contact info:
\noindent
\begin{minipage}[t][0.8cm][b]{0.33\textwidth} {\small name} \\ {\small address} \\ \small{post number}
\end{minipage}%
\begin{minipage}[t][0.8cm][b]{0.33\textwidth}\faMobilePhone \hspace{2.2mm} {\small666 66 666}\\ \faEnvelope \hspace{0.2mm} \small{email@.com}
\end{minipage}%
\begin{minipage}[t][0.8cm][b]{0.33\textwidth}\hfill {\small \today}
\end{minipage}
% thumb index:
\begin{tikzpicture}[remember picture,overlay]
\draw [color=color1, fill=color1]
([xshift=-35pt]current page.north east) rectangle (current page.south east);
\node[rotate=90,white,font=\Large\bfseries,anchor=south east]
at ([yshift=-10pt,xshift=-5pt]current page.north east)
{Application};
\end{tikzpicture}
\vspace{0.3cm}
\textbf{Application on the job} \\
\lipsum
\begin{tikzpicture}[remember picture,overlay]
\draw [color=color1, fill=color1]
([xshift=-30pt]current page.north east) rectangle (current page.south east);
\node[rotate=90,white,font=\Large\bfseries,anchor=south east]
at ([yshift=-10pt,xshift=-5pt]current page.north east)
{Appendix};
\end{tikzpicture}
\end{document}
The result:

Of couse, a new command will simplify the job:
\newcommand\ThumbIndex[1]{%
\begin{tikzpicture}[remember picture,overlay]
\draw [color=color1, fill=color1]
([xshift=-35pt]current page.north east) rectangle (current page.south east);
\node[rotate=90,white,font=\Large\bfseries,anchor=south east]
at ([yshift=-10pt,xshift=-5pt]current page.north east)
{#1};
\end{tikzpicture}%
}
And the, in the document,
\ThumbIndex{Application}
and
\ThumbIndex{Apprndix}
at the desired locations. The complete code:
\documentclass[a4paper,12pt,twoside]{article}
\usepackage[norsk]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[margin=2.5cm]{geometry}
\usepackage{setspace}
\onehalfspacing
\usepackage{parskip}
\usepackage{fontawesome}
\usepackage{graphicx,tikz}
\definecolor{color1}{rgb}{0.22,0.45,0.70} %blå fra cvmodern.
\usepackage{lipsum}
\newcommand\ThumbIndex[1]{%
\begin{tikzpicture}[remember picture,overlay]
\draw [color=color1, fill=color1]
([xshift=-35pt]current page.north east) rectangle (current page.south east);
\node[rotate=90,white,font=\Large\bfseries,anchor=south east]
at ([yshift=-10pt,xshift=-5pt]current page.north east)
{#1};
\end{tikzpicture}%
}
\begin{document}
\setcounter{page}{0}
\pagenumbering{roman}
%Contact info:
\noindent
\begin{minipage}[t][0.8cm][b]{0.33\textwidth} {\small name} \\ {\small address} \\ \small{post number}
\end{minipage}%
\begin{minipage}[t][0.8cm][b]{0.33\textwidth}\faMobilePhone \hspace{2.2mm} {\small666 66 666}\\ \faEnvelope \hspace{0.2mm} \small{email@.com}
\end{minipage}%
\begin{minipage}[t][0.8cm][b]{0.33\textwidth}\hfill {\small \today}
\end{minipage}
\ThumbIndex{Application};
\vspace{0.3cm}
\textbf{Application on the job} \\
\lipsum
\ThumbIndex{Appendix};
\end{document}
If you want to move the thumbindex, you can use another argument argument, or make the shift depend on sectional unit counters:
\newcommand\ThumbIndex[2]{%
\begin{tikzpicture}[remember picture,overlay]
\draw [color=color1, fill=color1]
([xshift=-35pt]current page.north east) rectangle (current page.south east);
\node[rotate=90,white,font=\Large\bfseries,anchor=south east]
at ([shift={#2}]current page.north east)
{#1};
\end{tikzpicture}%
}
And then
\ThumbIndex{Application}{(-5pt,-10pt)};
and
\ThumbIndex{Appendix}{(-5pt,-120pt)};
The result of the code below:

The complete code:
\documentclass[a4paper,12pt,twoside]{article}
\usepackage[norsk]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[margin=2.5cm]{geometry}
\usepackage{setspace}
\onehalfspacing
\usepackage{parskip}
\usepackage{fontawesome}
\usepackage{graphicx,tikz}
\definecolor{color1}{rgb}{0.22,0.45,0.70} %blå fra cvmodern.
\usepackage{lipsum}
\newcommand\ThumbIndex[2]{%
\begin{tikzpicture}[remember picture,overlay]
\draw [color=color1, fill=color1]
([xshift=-35pt]current page.north east) rectangle (current page.south east);
\node[rotate=90,white,font=\Large\bfseries,anchor=south east]
at ([shift={#2}]current page.north east)
{#1};
\end{tikzpicture}%
}
\begin{document}
\setcounter{page}{0}
\pagenumbering{roman}
%Contact info:
\noindent
\begin{minipage}[t][0.8cm][b]{0.33\textwidth} {\small name} \\ {\small address} \\ \small{post number}
\end{minipage}%
\begin{minipage}[t][0.8cm][b]{0.33\textwidth}\faMobilePhone \hspace{2.2mm} {\small666 66 666}\\ \faEnvelope \hspace{0.2mm} \small{email@.com}
\end{minipage}%
\begin{minipage}[t][0.8cm][b]{0.33\textwidth}\hfill {\small \today}
\end{minipage}
\ThumbIndex{Application}{(-5pt,-10pt)};
\vspace{0.3cm}
\textbf{Application on the job} \\
\lipsum
\ThumbIndex{Appendix}{(-5pt,-120pt)};
\end{document}
Remarks
Perhaps the background package could also be used here, but with the information in the question I cannot decide.
\small and similar switches don't receive arguments. Instead of \small{text} you should use {\small text}.
I added some % at the end of the first two minipages to prevent spurious blank spaces from causing overfull \hboxes.
;that follows the second argument of\ThumbIndexis not really part of how the command should be used. – jon Jul 31 '15 at 01:54