How much time would it take to write a C++ compiler using lex/yacc?
Where can I get started with it?
There are many parsing rules that cannot be parsed by a bison/yacc parser (for example, distinguishing between a declaration and a function call in some circumstances). Additionally sometimes the interpretation of tokens requires input from the parser, particularly in C++0x. The handling of the character sequence >> for example is crucially dependent on parsing context.
Those two tools are very poor choices for parsing C++ and you would have to put in a lot of special cases that escaped the basic framework those tools rely on in order to correctly parse C++. It would take you a long time, and even then your parser would likely have weird bugs.
yacc and bison are LALR(1) parser generators, which are not sophisticated enough to handle C++ effectively. As other people have pointed out, most C++ compilers now use a recursive descent parser, and several other answers have pointed at good solutions for writing your own.
C++ templates are no good for handling strings, even constant ones (though this may be fixed in C++0x, I haven't researched carefully), but if they were, you could pretty easily write a recursive descent parser in the C++ template language. I find that rather amusing.