Some info:
The 404 error means Tomcat doesn't have any file or servlet or anything exposed for serving at that URL path.
The 500 error means Tomcat has a servlet (or jsp) exposed at that URL path and an unexpected exception was generated while trying to run the java code.
Collection of things we know need to be changed:
$.get('/jirarequest/whatever','OK',function(op){
and
<servlet>
<servlet-name>JiraSyncServlet</servlet-name>
<servlet-class>JiraSyncServlet</servlet-class>
</servlet>
Plus you can test in the browser by typing this URL which should return OK:
http://localhost:8080/Jira-Synchronization/jirarequest/whatever?OK