"The hard working employee and I on payday" is not a sentence, it is a fragment.
There is no verb, so the main prescriptive rule about when to use forms like "I" vs. "me" don't really give a definite answer (it would say to use "me" in a sentence like "This is a picture of the hard working employee and me on payday", but "I" in a sentence like "The hard working employee and I on payday look like this").
In practice, native speakers tend to use "me" in fragments like this except for after "and", where many use "I". However, the use of "and I" outside of the subject of a sentence is widely regarded as a "common mistake" (as seen from comments here). The use of "I" in this context is assumed to have originated as a "hypercorrection" (basically, an overcompensation for a grammar rule taught in school: to avoid using "me" as part of the subject of a sentence), but descriptive linguists differ on how they explain its current use.
The question of whether to use "I" or "me" in this context was previously asked at photo caption ... me or I?. This was wrongly marked as a 'duplicate' of Should I put myself last? "me and my friends" vs. "my friends and me" or "my friends and I"