Well, it’s not a real sentence as has already been said. But what would he put if Sue hadn’t been there?
“Me at Yellowstone”, I suspect. In which case, he should use “me” for the picture with Sue, too. To make it sound better, put “Me and Sue at Yellowstone”.
Job’s a good 'un.
For me, the question is: What would you write, in one word, under a picture of yourself: “I” or “Me”? Myself, I would write “me”, just as I would say “It’s me.”
Let me check. The problem is that you’re not sure that he was actually at Yellowstone?
No, I’m sure he was there 
The problem is that he is assaulting my grammatic sensibilities.
This is a good question. A couple of years ago I picked up a copy of Bill Clinton’s new book and all the pictures have “I” in the caption: Mother and I at…, President Kennedy and I…, Monica and I in the…, etc.
Initially I thought that it was incorrect, but then thought there’s no way a writer, editor, and publisher could all be wrong, so I capitulated and thought I must be wrong.
Now I’m not sure again.
You could always go with Sue’s and my being at Yellowstone.
In these cases, there’s an implied verb. Where you see “Mother and I at…” read “Mother and I are at…” The first person pronoun is the subject of the sentence, so the use of I is correct.
A better test, rather than dropping the other person, is to replace “X and I” with “we”, “X and me” with “us”.
This gives us:
“Sue and I at Yellowstone” = “We at Yellowstone”
“Sue and me at Yellowstone” = “Us at Yellowstone”.
I would say that “me” is the right pronoun to use.