"I am down at noon."
Translation:Délben lent vagyok.
I'm not a native, but my gut instinct tells me that "vagyok" refers to whatever is directly before it. By saying "Délben vagyok lent" it sounds kind of like you're saying "I am [at noon]..." then just saying the word "lent." I don't think it's wrong, but it just sounds unnatural; it's like you're calling yourself "at noon."
What are you? You are "down/below," so that word needs to come just before "vagyok." So your most natural-sounding translations are "Délben lent vagyok" or "Lent vagyok délben." The first emphasizes "at noon," while the second just states when.
Actually, there is another use, not necessarily downstairs. To go from a major city to a smaller settlement or to somewhere in the countryside, or even from anywhere to a location to the south, you can use "lemenni" - to go down. Most significantly, but not exclusively, it is used for going from the capital, Budapest, to anywhere else in the country. Let's say we are in Budapest and we want to go to Lake Balaton. Let's go to Lake Balaton: "Menjünk (le) a Balatonra". And, if I leave home at 10 am, I am (down) there at noon: "Délben lent vagyok".
And the opposite way, to go to a major center, you can use "felmenni", to go up. Tomorrow I am going (up) to Budapest: "Holnap felmegyek Budapestre".