Here's one possible solution:
\documentclass{article}
\usepackage{amsthm}
\usepackage{thmtools}
\usepackage{xstring}
\usepackage{etoolbox}
\newcommand\ExtInf{%
\StrBetween[1,2]{\sectiontitle}{ }{ }[\pmonth]
\StrBefore{\sectiontitle}{ }[\ptitle]
\StrBehind[2]{\sectiontitle}{ }[\pyear]%
}
\declaretheoremstyle[
spaceabove=\topsep, spacebelow=\topsep,
headfont=\normalfont\bfseries,
bodyfont=\normalfont\itshape,
postheadspace=0em,
headpunct=,
headformat=\pyear.\NUMBER~(\pmonth)~(\ptitle):~,
]{mystyle}
\declaretheorem[style=mystyle,name=]{problem}
\makeatletter
\@addtoreset{problem}{section}
\pretocmd{\@sect}{\gdef\sectiontitle{#7}}{}{}
\apptocmd{\@sect}{\ExtInf}{}{}
\makeatother
\begin{document}
\section{Ph.D. August 2004}
\begin{problem}
Test problem.
\end{problem}
\begin{problem}
Test problem.
\end{problem}
\section{M.Sc. September 2010}
\begin{problem}
Test problem.
\end{problem}
\end{document}

Some explanations:
The etoolbox packagewas used to capture the title of each section in the macro \sectiontitle.
The xstring package was used to extract from the string stored in \sectiontitle the relevant sub-strings; these sub-strings are: the degree, the month, and the year, and they are stored in \ptitle, \pmonth, and \pyear, respectively.
The macro \ExtInf wraps this process, and etoolbox is used so that \ExtInf is invoked with every new \section.
The title for each section is expected in the form <degree><space><month><space><year>
The thmtools package was used as a front-end to amsthm; with the help of this package, the theorem-like structure problem was defined; using the headformat key, the information about the year, number, month and degree are automatically included with the chosen formatting.
A second simplified version:
This time there's no need to extract substrings; the \ExtInf macro receives three mandatory arguments:
\ExtInf{<title>}{<month>}{<year>}
stores the information in the macros \ptitle, \pmonth, and \pyear and also creates the corresponding \section using the macros; then, thmtools uses those macros in the headformat key to give the desired formatting for the heading of the problem environment:
\documentclass{article}
\usepackage{amsthm}
\usepackage{thmtools}
\usepackage{etoolbox}
\newcommand\ExtInf[3]{%
\gdef\ptitle{#1}
\gdef\pmonth{#2}
\gdef\pyear{#3}
\section{#1~#2~#3}
}
\declaretheoremstyle[
spaceabove=\topsep, spacebelow=\topsep,
headfont=\normalfont\bfseries,
bodyfont=\normalfont\itshape,
postheadspace=0em,
headpunct=,
headformat=\pyear.\NUMBER~(\pmonth)~(\ptitle):~,
]{mystyle}
\declaretheorem[style=mystyle,name=]{problem}
\makeatletter
\@addtoreset{problem}{section}
\makeatother
\begin{document}
\ExtInf{Ph.D.}{August}{2004}
\begin{problem}
Test problem.
\end{problem}
\begin{problem}
Test problem.
\end{problem}
\ExtInf{M.Sc.}{September}{2010}
\begin{problem}
Test problem.
\end{problem}
\end{document}
