Learn Italian: Prego, l'ufficio è aperto. - Please, the office is open.

Prego is a very versatile word! While it is the standard reply to grazie (thank you), it is also commonly used as a polite invitation. In this context, it means 'please go ahead,' 'come in,' or 'after you.' You will hear it a lot in shops, restaurants, and offices.

Is this sentence using the formal register (dare del Lei)?

Because this sentence doesn't actually use the word 'you' or a verb tied to 'you', it doesn't strictly contain the formal Lei grammar. However, using prego to invite someone in is exactly the kind of polite language you use when speaking formally to a client, customer, or someone you don't know well.

Why is it l'ufficio instead of il ufficio?

Ufficio is a masculine singular noun that starts with a vowel. In Italian, masculine singular words that start with a vowel take the article l'. It is basically a shortened version of lo (so lo ufficio becomes l'ufficio) so the two vowel sounds don't crash into each other when speaking.

Does the word aperto change depending on what is open?