Veins carry oxygen-poor blood cells back to the heart and lungs to be re-oxygenated, so it is reduced blood flow in the arteries that can cause a drop in sensation in the legs, arms, hands, etc. Arteries contain freshly oxygenated blood and carry it throughout the body. As you mentioned, it is the capillaries that directly provide the nutrients to nerve cells which in turn allows for signal transduction to occur, but it is the arteries that supply the capillaries with fresh blood.
Nerve cells require a continuous flow of nutrients, especially Ca2+ and Na+ ions in order to properly function. As you may recall, nerve signal transduction depends on the nerve cell being able to route a wave of depolarization across the length of the axon. Without the vital nutrients and oxygen that fresh blood supplies, the electrical gradient can't be produced, resulting in no signal being sent to the brain. Your body interprets this lack of signal as numbness. You can try this out on your own by pressing against the large artery on the inside of your upper arm. After a few minutes you'll notice a drop in the sensation of your entire limb, which is due to your nerve cells not having fresh nutrients.
As for veins, blocking a vein will only lower the amount of blood that is being re-oxygenated, but there are hundreds of veins in your body that can compensate for a single vein being blocked, so it is very unlikely that blocking a single vein will have any effect. The only way that vein blockage could cause problem is if it is completely blocked off, resulting in a buildup of oxygen-poor blood not being able to leave a site, at which point numbness will be the least of your worries, and gas gangrene and swelling will become more pressing issues.