Either A) the debris is so dense that you can laser-broom it out of orbit or B) it is sufficiently sparse that you can work on cladding your spacecraft in suitable protective layers and you'll be OK. If the debris is so small, multilayer thin Whipple shields seems like they should be more than adequate.
Remember that everyone hoos and hahs about the velocity term in the kinetic energy equation, $e_k = \frac{mv^2}{2}$, and relative orbital velocities in a collision might end up as high as 20km/s or more, but your submillimetre dust grains are gonna end up with millgram and microgram masses. A milligram at 20km/s only carries 200J of kinetic energy... that's less than a 9mm bullet, and it doesn't have nearly the amount of momentum that the 9mm will have. Most collisions will be less energetic than that, though, especially if most of your debris is going in the same direction.
A few thin sheets of aluminium and maybe some slabs of dense polyethelene with a decent amount of spacing seems like it'd do an excellent job of protection. So much so, that you could imagine throwing up some dust-absorbers on sub-orbital trajectories whose only job is to spend a little time at orbital heights, soaking up debris in their layers of UHMWPE before falling back to Earth. You don't even need a serious rocket for that; suborbital rocketry is some orders of magnitude cheaper than the real thing, after all. A big dumb payload in a real orbital rocket can do a better job, unfurling a nice big umbrella of tough plastic and aerogel to soak up the worse junk without breaking up, then ditching back into the atmosphere before the wear gets serious enough to risk the shield breaking up.
sub millimeter debris, undetectable until you get hit
If it is dense enough to be dangerous, you can see it by the effect it will have on light transmission and reflection. You'll be able to see it alright... after all, we can see the interplanetary dust cloud in space around Earth, and that's got a density of about 1 fleck of dust per million cubic metres because the cloud is big enough to affect light transmission.
I'm not even sure if it would be possible to have a debris field with such a high density that hundreds of hits by debris would be basically guaranteed,
They might not be guaranteed instantly, but a big cloud of crud in a slightly different orbit is going to hit you hard at regular intervals. It'll grind you down over time, certainly.
but I'm decently sure that if all the debris is moving in one general direction, it should be possible.
If you want to maximise kinetic energy, and hence the hazard presented by your debris, it needs to be in a mix of orbits. Eccentric ones give you higher velocities but reduce the chances of intersection in any given orbit, but to maximise damage you'll be wanting debris orbiting in totally different directions.
When everything is travelling in more or less the same direction, the risk of damage can be much reduced by injecting yourself into an orbit that minimises relative velocities. You'll not be getting 20km/s impactors in that situation, that's for sure.