Suppose I have a picture of me hanging out with Bob, and I want to add a caption to it. Is one of the following more correct than the other?
- "Bob and me"
- "Bob and I"
More generally, do grammar rules apply to sentence fragments at all?
As you say, it's a sentence fragment. It doesn't really matter which you choose, the choice is more a matter of style, than of correctness.
But we can look at what would be usual.
If you change the context, and say you're trying to identify an unknown person, and you have a bunch of photos. Would you say *"Here's another one of bob and he."? I certainly wouldn't.
So since your fragment is probably an elision of "This is a photo of Bob and me.", it's clear one would most often use an object, not a subject.