The phrase "I am just about" in this context means that the speaker is almost or nearly the only thing that arrived in one piece at the Trade Fair in Frankfurt. It suggests that most other things or items may not have arrived intact or undamaged. The use of "just about" indicates a degree of uncertainty or approximation, emphasizing that the speaker's arrival in one piece is somewhat unexpected or unusual given the circumstances.

In my opinion this is a very authentic example of what a person might naturally say, and because it is 'natural' for native speakers to say this type of sentence they might not think too much about the grammar. Therefore it might be useful to think about this as a typical idiom or phrase, rather than trying to analyse the grammar. In this example "I am just about the only thing that arrived here in one piece." = "Nearly everything else did not arrive here or if it did arrive it was broken". The person is probably exaggerating a bit. It might be more logical to say "I am about the only thing that arrived here in one piece." and omit the word "just" completely, but we tend to add extra words to sentences when we want to emphasise a point, especially when speaking because we can stress the pronunciation of the extra word to convey the strength of feeling that we want to communicate.