There is no direct link between the capillaries in the sinuses and the ear.
The Eustachian tubes drain the middle ear (between the eardrum and the inner ear) into the nasopharynx, the part of the throat that is just behind the nose.
The para-nasal sinuses drain into the nasal passages themselves at different points. The sinuses and middle ear constantly produce some fluid, which is either reabsorbed or drained.
Often what happens is that in the case of a common cold or other upper respiratory tract infection, the nasopharynx and nasal mucosa are both infected. The inflammation causes swelling and blocks off the Eustachian tubes and the drainage pathways of the sinuses and the trapped fluid itself causes sinus & ear pain and can get infected causing sinusitis and otitis media. Alternatively a viral infection can also go up into the middle ear and sinuses causing direct infection.
The process that occurs when you go on a plane is slightly different. The Eustachian tubes and sinus openings collapse due to pressure and thus the natural fluid that is produced accumulates as well as there being an inability to neutralise the internal and external pressure (thus why making your ears 'pop' works- you force the Eustachian tubes open via positive pressure)
Regarding blood supply of the paranasal sinuses and the ear:
The ear is supplied by the posterior auricular and superficial temporal arteries of the external carotid artery.
The paranasal sinuses are supplied by various branches of the internal and external carotid artery including the maxillary artery, anterior and posterior ethmoidal arteries, ophthalmic artery, supraorbital and supratrochlear arteries