According to Aulus Gellius, here is the epitaph of Pacuvius (Gellius I.24.4)

“Young man, even though you hurry by, this stone
asks you to look on it and then to read what is written.
Here is where you find interred the bones of the poet
Marcus Pacuvius. I desire that you know this. Farewell.”

Adulescens, tam etsi properas te hoc saxum rogat
Ut sese aspicias, deinde quod scriptum est legas.
Hic sunt poetae Pacuvi Marci sita
Ossa. Hoc volebam nescius ne esses. Vale.

Literary–both fictionalized and not–epigraphs were part of the Greek literary tradition at least to the 6th century BCE. From the 5th century, we have Simonides’ epitaph at Thermopylae:

“Stranger, go tell the Spartans that we lie here
obedient to their commands.”

Ω ξεῖν’, ἀγγέλλειν Λακεδαιμονίοις, ὅτι τῇδε
κείμεθα τοῖς κείνων ῥήμασι πειθόμενοι.

This epigram seems ‘real’ enough, but during the Hellenistic period, poets like Callimachus seem to have made a game of composing funerary epigrams. Here’s one he wrote about himself (or not):

Callimachus, epigram 21.

“Whoever you are lifting your foot near my grave
Know that I am the child and father both of Cyrenian Callimachus.
You would know both men. One led the soldiers of his country,
And the other sang songs beyond envy.
Don’t be surprised: whomever the Muses behold at birth
Are not abandoned friends as they grow grey.”

῞Οστις ἐμὸν παρὰ σῆμα φέρεις πόδα, Καλλιμάχου με
ἴσθι Κυρηναίου παῖδά τε καὶ γενέτην.
εἰδείης δ’ ἄμφω κεν• ὁ μέν κοτε πατρίδος ὅπλων
ἦρξεν, ὁ δ’ ἤεισεν κρέσσονα βασκανίης.
[οὐ νέμεσις• Μοῦσαι γὰρ ὅσους ἴδον ὄμματι παῖδας
†ἄχρι βίου† πολιοὺς οὐκ ἀπέθεντο φίλους.]

Gellius also adds to Pacuvius’, an epitaph of a more commonly known comedian, Plautus:

 
“Now that Plautus has found death, Comedy weeps,
Abandoned on the stage. And then, Laughter, Play and Jest
mourn together with all the uncountable Measures.”

postquam est mortem aptus Plautus, Comoedia luget,
scaena est deserta, dein Risus, Ludus Iocusque
et Numeri innumeri simul omnes conlacrimarunt,

But despite all this weight and seriousness, I think that Naevius’ epitaph (also reported by Gellius) is the best:

“If it were right for gods to mourn for mortals
Then the Muses would mourn the poet Naevius.
And when he was brought down to death’s warehouse
Rome would forget how to speak the Latin tongue.”

Immortales mortales si foret fas fiere
Fierent divae Camenae Naevium poetam
Itaque postquamst Orchi traditus thesauro
Obliti sunt Romae loquier lingua latina.