"He lives in an important palace."
Translation:Lui vive in un palazzo importante.
In this case it's easy: "nel" is a compound of in+il, so "nel un palazzo" would be "in the a palace". See http://italian.about.com/library/fare/blfare153a.htm
The above translation is wrong. "Palazzo" is a word that's also used in English when referring to Italian palazzos that were usually built by princes from the 1400's on and which were converted to apartment buildings at the end of the royal period. I grew up part time in a palazzo in Naples ...but I would never translate that as growing up in a "palace" in English. -- I would say "I grew up in a palazzo" using the Italian which Italians know means my family didn't own the palazzo but that instead I grew up in one of the apartments there. Palazzo has a very different meaning than 'palace' by our terms.