Non-Newtonian basically means that you can't use it in an hourglass, because the flow is unpredictable (for today's fluid mechanics). I kind of like the idea though.

Non-newtonian fluids fall into several categories. [placid_turmoil] will fill your veins if you try to model these fluids using Newton's original force balance on a differential element of fluid:

t = u* (dv/dy)

This simple equation tells us that if a force, t, is exerted on a fluid parallel to it's surface, then the perpendicular velocity gradient, (dv/dy) will be constant. In non-newtonian fluids, this velocity gradient will not be constant, but will be dependent on a variable viscosity, u. The proportionality constant, u, is not constant at all in non-newtonian fluids. Instead, it depends on temperature, time, pressure, position in the fluid, etc. These variations can sometimes be successfully modeled, and are therefore (painstakingly) predictable, contrary to previous annotations.

The generalized non-newtonian equation is as follows:
t = u*(dv/dy)
u = f(t,T,P,y,time)

Notice how t is dependent on u and u is dependent on t. This computation feedback is essentially what drives the mathematicians mad. Finite elemental analysis can take care of most the trickery, but it takes along time to model anything significant.

Ketchup is a shear thinning fluid, which means that it flows easier the greater the applied force (u ~ 1/t). Small forces do nothing to it. I can't reliably predict if this would work just fine, since a force balance at the bottleneck needs to account for gravity and the fact that very little shear is transmitted near the walls of a surface. I think it would be a terribly slow device, really. Also, any usage of ketchup is going to foil your timing, and flowrate is going to depend drastically with the amount left in the top, also leading to terribly confusing conversion factors and such, if any ketchup flowed through at all.