You are correct: me is the object pronoun and should be used here, since it is going in the object position.
Normally you would probably say "keep us updated", so "keep him and me updated" may feel a bit off even though it's perfectly correct.
As to the people who tell you that you should always use "I" in conjunction with a name: they are WRONG, WRONG, WRONG! Ignore them! It's a form of "hypercorrection"--when you "correct" something that's already right based on some rule that doesn't actually apply.
Too many people have gone from:
"you never say 'Tom and me went to the store' (because "Tom and me" is the subject, so you need I instead of me)"
to
"You never say 'Tom and me', ever"
and then had to invent a reason for their mis-remembered rule.
Let's review. What you are trying to express is:
John lost his friend. John was crying for his friend.
Of course, such repetition is undesirable, so English has a tool you can use, called a relative-pronoun. Relative pronouns include who, whom, that, and which. Of those choices, who and whom can be used to refer to people. Who is used for subjects; whom is used for objects.
You probably know all that already, but just need advice in applying the rules.
The pronoun you want is the one in the objective case (whom), because John lost his friend. (Using the subjective case pronoun who would mean that the friend lost John, which is not what you want.)
Remember, also, that relative pronouns help you refer to a noun that would otherwise be repeated. Relative pronouns don't stand alone. The correct sentence should be:
John was crying for the friend whom he lost.
That said, even native English speakers often fail to distinguish between subjective and objective cases correctly. Therefore, you have to be careful about parroting patterns that you see and hear, as you may be copying incorrect examples. On the other hand, you are also unlikely to be stigmatized for choosing the wrong case in everyday speech.
Best Answer
What role does the pronoun play in the sentence? If it's a subject, it should be in the subjective case. If it's an object, it should be in the objective case.
As I parse this sentence, "him" is the object of the preposition "but". The prepositional phrase "but him" modifies the pronoun "nobody". The nominative phrase "nobody but him" is the subject of the clause. "Was" is the verb. "Present" is a predicate adjective.
However, there is an alternate view. The word "but" can also be a conjunction. As a conjunction, it would join the subject "nobody" with the subject "he". As a subject, "he" takes the subjective case.
Since both options are grammatically possible, which should we prefer?
I could say that, in my dialect, the first sounds more natural. That may be true, but it isn't very helpful.
There is a version of this sentence such that "but" must be a preposition: "Nobody was present but him." To my ear, this version means the same thing as the original. For that reason, I recommend treating this instance of "but" as a preposition, and treating "him" as the object of a preposition.