Right align inside header

Don't load fancyheadings, which is the obsolete version of fancyhdr.

The easiest way is to use tabular which will take care of the alignment:

\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhf{} % clear all fields
\fancyhead[R]{%
  \scshape
  \begin{tabular}[t]{@{}r@{}}
  Tutorial\\The periodic table
  \end{tabular}}
\fancyhead[L]{%
  \scshape
  \begin{tabular}[t]{@{}r@{}}
  My city\\University of somewhere
  \end{tabular}}
\fancyfoot[C]{\thepage}

Set the \headheight to the value that fancyhdr will show you in the .log file. For a 10pt document it says to do

\setlength{\headheight}{24pt}

You may typeset the left header \raggedleft inside a minipage with a width of the widest element (calculated with the help of \widthof, which is provided by the calc package).

\documentclass{article}
\usepackage{fancyhdr}
\usepackage{calc}

\pagestyle{fancy}
\rhead{\textsc{Tutorial\\The periodic table}}
\lhead{%
  \begin{minipage}[b]{\widthof{\textsc{University of somewhere}}}
  \raggedleft
  \textsc{My city\\University of somewhere}%
  \end{minipage}
}
\cfoot{\thepage}

\begin{document}
    hello
\end{document}

EDIT: As pointed out by egreg in chat, one may replace the minipage with a varwidth environment "whose resulting width is the natural width of its contents" (package manual).

\documentclass{article}
\usepackage{fancyhdr}
\usepackage{varwidth}

\pagestyle{fancy}
\rhead{\textsc{Tutorial\\The periodic table}}
\lhead{%
  \begin{varwidth}[b]{10cm}
  \raggedleft
  \textsc{My city\\University of somewhere}%
  \end{varwidth}
}
\cfoot{\thepage}

\begin{document}
    hello
\end{document}

Output for both examples:

You can box the contents of the widest element and use this to width-adjust the shorter one:

\documentclass{article}
\usepackage{fancyhdr}% http://ctan.org/pkg/fancyhdr
\newsavebox{\myheadbox}% Heading storage box

\pagestyle{fancy}
\rhead{\textsc{Tutorial\\The periodic table}}
\lhead{\textsc{\savebox{\myheadbox}{University of somewhere}\makebox[\wd\myheadbox][r]{My city}\\\usebox{\myheadbox}}}
\cfoot{\thepage}

\begin{document}
    hello
\end{document}​

\myheadbox contains the "University of somewhere" (in the appropriate font, \scshape). Then, "My city" is right-aligned in a \makebox of width \wd\myheadbox (width of \myheadbox).