inertial pressure on in a rotating syringe

The way I see it this problem is similar the problem of column of water and the pressure rising due to gravity pulling down on the liquid - in which case pressure, $P$, is given by

$P = h rho g$

where $h$ is the height of liquid, $rho$ is the density and $g$ the acceleration due to gravity - but you probably knew this already.

The point I want to make here is that the height of water is important because the forces all add up from top down to bottom (so the pressure at the bottom of the oceans can be very very high - 100s or 1000s or more atmospheres)

Here the issue is the that the force (and hence acceleration) due to the rotation changes - as the radius gets larger the force gets bigger of course.

So calculate the pressure at $r1$ and $r2$ you need to do an integration to add up all the forces for the liquid 'above' $r1$ and $r2$ - so on your diagram you need the height (edit - ah not labelled, but that it is in writing on the diagram.)

The good news is that the formulae for $r1$ and $r2$ will be the same as the cross sectional area of the tube does not matter - but again you need to know the height of liquid - or length of liquid towards the centre.

Finally, Centripetal force = centrifugal - action and reaction... here you can consider the centrepetal force is the same as the force the centrifugal force. Think of it this way, the pressure at $r1$ or $r2$ must provide the inward centrepetal force on the column of liquid above it to make it move in circular motion.