They are all the same,except the last one "should be arriving" suggests that there may be a change.The time he is "supposed to arrive" is 6 but there may be some problem.The others are all the same,just stating his time of arrival.

"He should" implies a degree of uncertainty, and in a formal letter, in a way it discredits you. I'd say "He will be arriving" is the most suited to a formal letter, and "He is arriving" best in an informal manner.

"He is arriving at the airport on Saturday at 6 in the morning," is the incorrect use of present tense to describe a future action. "He will arrive at the airport on Saturday at 6 in the morning," is a statement of fact. "He will be arriving at the airport on Saturday at 6 in the morning," suggests that something else will happen, like "he will be arriving just as it starts raining." "He should be arriving at the airport on Saturday at 6 in the morning," means I'm not sure if he will actually be arriving - it suggests doubt.

The first 3 mean the same thing, that he will definitely be arriving at the time given, but the last sentence, by using the word "should", it implies that that's the time he's supposed to be there, but he may or may not get there at that time.

By-the-way, if this is formal writing, spell out "six", unless you're being precise as in "6:00 AM".