if you can upload a JSP file you may print out some info like in this example: bestdesigns.co.in/blog/check-jsp-tomcat-version
Save this code into a file called tomcat_version.jsp:
Tomcat Version : <%= application.getServerInfo() %><br>
Servlet Specification Version :
<%= application.getMajorVersion() %>.<%= application.getMinorVersion() %> <br>
JSP version :
<%=JspFactory.getDefaultFactory().getEngineInfo().getSpecificationVersion() %><br>
When you access, http://example.com/tomcat_version.jsp, the output should look similar to:
Tomcat Version : Apache Tomcat/5.5.25
Servlet Specification Version : 2.4
JSP version: 2.0