plotyy works by having two axes on top of each other. In your above, the output handle AX will be a 1x2 vector. When you set gca to have new XTickLabels only the current axes is updated and not both. So what you need to do is turn off the xticklabels for the first axes and set them for the second (or vice versa).
E.g:
[AX,H1,H2]= plotyy(1:10,1:10,1:10,exp(1:10),'plot');
set(AX(1),'XTickLabel',[]);
set(AX(2),'XTickLabel',{'0-5','6-11','12-17','18-23','24-30','31-36','37-42','43-48','49-54','55-60','61-69'},'XTick',1:10,'FontWeight','bold','FontSize',20);