Learn English – When should I use “fan” and “admirer”

In grammatical terms, their is a possessive determiner, just like my, your, his, her, its and our. It indicates that what is described in the following noun phrase belongs to, or is in some other way associated with, the person or thing to which the possessive determiner refers. Just as my car is a car that belongs to me, so their house is a house that belongs to them.

There is a word with many uses. In your example, There are lots of skyscrapers in Dubai, it is what is sometimes called a ‘dummy subject’. It would be grammatical to say Lots of skyscrapers are in Dubai, but, generally, we don’t. We use there are (or there is) where French uses il est or il y a, German uses es ist, es sind or es gibt and Spanish uses hay.

Personally in the sentence means individually. It's possible to meet someone as part of a group, or be introduced to someone where you don't have an opportunity to really talk to them, but meeting someone personally means you introduce yourself and have an initial conversation.

In person means not via the phone, Internet, etc.. It means you and the person you are meeting are physically in the same place.

If the young man has never seen the father at all yet, really the distinction above is not needed. So merely say:

I have not had the honor of meeting you yet, but I hope see you soon.

Otherwise use personally. Unless you've only talked on the phone to the father and are finally actually going to see the father face-to-face.