Munkres is a very popular textbook, and google will find many sets of solutions to exercises available on the net.
Here are a few links, but note that they come with no authorization and do indeed contain some errors:
http://dbfin.com/2010/11/2000-munkres-topology-solutions-chapter-1/ (Other chapters by an obvious change)
http://www.math.cornell.edu/~matsumura/math4530/math4530web.html
http://www.math.cornell.edu/~erin/topology/munkres.pdf
http://www.math.lsu.edu/~lither/7510fall06/answers.pdf
http://www.math.ku.dk/~moller/e03/3gt/3gt.html
Wiki-like pages with loads of useful information:
http://topospaces.subwiki.org/wiki/Main_Page
http://planetmath.org/