How to add patterns inside shapes bounded by curves?

You need a closed shape to use patterns or fill. Shapes bounded by curves can be converted or redrawn like this:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{patterns}
\begin{document}
\begin{tikzpicture}
\draw[pattern=horizontal lines] (-5.5,1) arc (180:360:1 and .25) arc (180:360:2.5) arc (180:360:1 and .25) arc (0:-180:4.5) -- cycle;
\end{tikzpicture}
\end{document}

\documentclass{article}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
\draw  (-1,1) node (v1) {} circle (4.5);
\draw  (v1) circle (2.5);

\fill[red](-5.5,1) arc (180:360:1 and .25)
arc(180:0:2.5)
arc (180:360:1 and .25)
arc (0:180:4.5) ;
\end{tikzpicture}
\end{document}