Pope presumes the reader's acceptance of three facts about Horace: that he was fat; and short; and coughed. That he was fat and short he himself admits (Epistles I.20.24; Epistles I.4.15); the cough is puzzling. In a note to the Twickenham edition, John Butt writes that Horace refers to his cough in Satires I.9.32. Untrue; in that passage, Horace recalls a prophecy made when he was a boy that he could not be done in by poison, the sword, pleurisy, cough or the gout, but destroyed by a prattler. [p.97]Haynes thinks Pope has 'conjured an asthmatic Horace in order to be able to share another trait with him.' But this doesn't make much sense.
I've another theory. When Pope writes 'I cough like Horace', he means (and intends us to realise that he means) that his -- Pope's -- cough sounds, onomatopeically, like the name 'Horace'. A heavily aspirated initial wheeze, followed by the sibilant release of the cough itself. It's a word-game, a gag, and leads on to the two facts about the poet.