Heritage is one of Heaney's major themes across his body of work. The theme of heritage is taken up with irony and complexity in "The First Kingdom" as the poem draws an extended metaphor comparing rustic, rural life to that of a kingdom.
The suggestion that anchors the poem is a metaphor that presents pre-modern farm life as a kingdom where the domain of the royal family is a farm. This notion is very clear in the opening stanza.
"The royal roads were cow paths
[...] With seasoned sticks the nobles
lorded it over the hindquarters of cattle."
Being royal, in this context, carries with it no glamour but instead poses the life of the nobility as one of hard work and toil. Instead of playing a harp in her leisure, the queen works to milk the cows.
The central metaphor of the poem is extended through the first two stanzas and acts as a context for the entire poem. Thus the metaphor comparing life on the farm to the life of royalty in a kingdom can be called a conceit.
The individual metaphors comparing roads to cow paths and comparing milking cows to playing a harp are sometimes direct and sometimes implied.
The notion of heritage comes into play overtly in the last stanza, only after the content of what would be inherited has been examined. Standing to claim a legacy of brutal and unforgiving life that was/is nonetheless lived with full effort and exactitude, the poem's narrator expresses a diffident acknowledgement of family history (history as a series of normal, decidedly un-glamorous happenings).
There are no great names, it would seem, to lay claim to. There is only a continuation of toil and hard living that offers some pride but which is "demeaned" even as it remains "pious."
In writing about the poem, the underlying irony the animates the poem's conceit is a good place to start. Seeking to ennoble one's own family history in rural Ireland is to make a strange comparison, the poem suggests, essentially forcing the analogy of toil to nobility.