Cabin altitude on most airliners is typically 8000 feet.
Problems with blocked sinuses typically result in painful ears, as the pressure may be unable to equalise. This happens on descent; the Eustachian tubes need to be physically opened for the air to equalise. During climb it's usually not a problem - the shape of the apertures allows the air out, a bit like a non-return valve.
Physically opening the Eustachian tubes to balance the pressure between the (inner) ear and the sinus is done by swallowing. In an unobstructed (and normal) system, that provides sufficient movement for the air to get through. When the sinus is blocked it may be necessary to pinch the nostrils closed and blow - not too hard. If that doesn't work you're in a spot of bother. A painful inner ear will ensue. A ruptured eardrum is the worst case scenario.
The trick is to clear them deliberately and frequently on the way down, before it gets to the point of painful ears. A difference in air pressure tends to hold the tubes closed, thus requiring a greater physical movement than swallowing might provide.
I'd strongly recommend a nasal decongestant, or a spray bottle with a saline solution in it, to prevent the mucus becoming too 'gunky', and therefore more likely to form a blockage that can't be moved. Use it regularly during flight. Air in the cabin is dry. Some people have reported success with hayfever medications designed to dry the sinuses out.