As others have suggested, it's crucial to update your TeX distribution first (some people would even say: don't even think about asking a question without having updated your distribution first).
As you seem to be using XeLaTeX, loading fonts via packages made for pdfTeX isn't the best option in most cases. You should really be using fontspec
IMHO.
I'm sure your operating system already includes Times New Roman -- not necessarily under that name, though, and not necessarily a version that includes Cyrillic. One version that definitely does is the one shipped with Windows. Here's how to install it, in case it's not available on your machine yet. Once it is, something like this should work without any problems (unless something more fundamental is wrong with your TeX system):
\documentclass{scrartcl}
\usepackage{fontspec}
\setmainfont{Times New Roman}
\listfiles
\begin{document}
Рома́н О́сипович Якобсо́н — российский и американский лингвист и литературовед,
один из крупнейших лингвистов XX века, оказавший влияние на развитие гуманитарных
наук не только своими новаторскими идеями, но и активной организаторской деятельностью.
\end{document}

A few other Cyrillic-aware fonts that can be used with fontspec
are listed here.
\renewcommand{\rmdefault}{ptm}
.\usepackage{times}
doesn't work. You may be luckier with\usepackage{paratype}
. The font isn't Times, though. See paratype.com