So after doing a little more digging through the 'plot' documentation, I've come up with this:
h=initial(sys,x0,t);
ix=find(h<=m,1);
initial(sys,x0,t)
hold on
% plot horizontal line, vertical line and marker
plot([0 t(ix)],[m m],'b--',[t(ix) t(ix)],[0 m],'b--',...
t(ix),0.02*(pi/2),'ro','MarkerSize',5,'MarkerFaceColor','r')
Is this a good solution? I feel it's kind-of "dirty", if you know what I mean, but it gets the job done. If anyone has any other suggestions, you're more than welcome to reply!