Is it normal to feel like pressure is building behind my eye whenever I tilt my head downward?

Eye pressure. No, not normally. Could be a variety of possibilities including but not limited to sinus problems, tension headache, ocular inflammation, high eye pressure, or increased brain pressure. A comprehensive eye exam can help elucidate the diagnosis.
No. This sounds like sinus disease. Best to get the sinuses checked out and if the sensation is persistent, check the blood vessels and orbits.