The first thing that strikes me about this poem is that I don't know how "Arbuthnot" is to be pronounced. ARbuthnot? ArBUTHnot? Is the "not" even fully pronounced or is the "o" sort of squeezed out, swallowed, as if it were "n't"? I do not know, and these things bother me, in this case every time I think about the poem.

But I have carried on reading it in spite of that glitch.

Arbuthnot was a well-known figure in the intellectual circles in which Pope moved, and also physician and close personal friend of Pope--see his Wikipedia entry for more information. It was mainly in his capacity as friend and literary sympathizer that Pope addressed this poem to him. I should perhaps emphasize "sympathizer," because the poem is all about Pope, not Arbuthnot, and is basically a 419-line complaint. 

(I just noticed that though the poem is in heroic couplets, 419 is an odd number. Something is wrong here. Is there an unrhymed line in there somewhere? A triple rhyme? Or is the line numbering wrong? That seems unlikely, as I have the poem in an anthology and also in a Best of Pope volume, and the line count is the same in both. Perhaps I'll investigate further. Or perhaps not.)

What is Pope complaining about? Initially, the crowd of litterateurs of the second rank or lower who want something from him--his criticism, which will be followed by requests for his assistance; his influence with publishers or theater managers; sometimes also or instead, his money. 

Fire in each eye, and papers in each hand,
They rave, recite, and madden round the land.

They invade his residence at Twickenham:

All fly to Twit'nam, and in humble strain
Apply to me, to keep them mad or vain.

A long list of the pests, their particular entreaties, and Pope's unhappy and witty responses follows.  Who is worse, the critics who denounce his work, or the poets who want him to read theirs?

A dire dilemma! either way I’m sped,             [i.e. to the grave]
If foes, they write, if friends, they read me dead.
Seized and tied down to judge, how wretched I!
Who can’t be silent, and who will not lie.
To laugh, were want of goodness and of grace,
And to be grave, exceeds all power of face.
I sit with sad civility, I read
With honest anguish, and an aching head;

Less witty and more venomous is what follows: the settling of scores with literary enemies who have injured and insulted him. Some of these are explicitly named, some masked behind a classical reference, like this one:  

Let Sporus tremble—What? that thing of silk,
Sporus, that mere white curd of ass’s milk...?

And that's only the beginning of a dozen or more lines vilifying Sporus. This is pretty harsh, whoever "Sporus" is meant to be, but downright vicious once you've been informed by a footnote that (1) Sporus was a boy, "Nero's favorite sexual partner" and (2) the reference was meant for Lord Hervey, confidante of Queen Caroline. I wonder that some sort of action, legal or other, was not taken against Pope; perhaps because there was too much truth in the characterization? (Here's Lord Hervey's Wikipedia entry, for context if not definite judgment.)

The contemporary non-academic reader--myself, for instance--is probably not very interested in the quarrels themselves, the personalities or the substance. Little of this would be of anything other than historical interest apart from the quality of the writing--the skill and power of expression, the virtuosic handling of the couplet, whether in epigrammatic pairs or sustained thought over a dozen or more lines. Only if one is interested enough in 18th century literature to be interested in Joseph Addison does one care that "Atticus," who gets twenty or so lines of snark for his domination and manipulation of a literary circle which did not include Pope, is Addison, who at one time had been Pope's friend. That information is not required for appreciation of the scorn in which Pope holds a critic who will

Damn with faint praise, assent with civil leer,
And without sneering, teach the rest to sneer;
Willing to wound, and yet afraid to strike,
Just hint a fault, and hesitate dislike.

"Damn with faint praise," like many trenchant bits from Pope, has passed into broad usage. I find myself doing it more often than I would like, and have been on the unhappy receiving end of it a few times (though as far as I know never from malice, but rather kindness bordering on pity, which may feel worse). 

Pope's work is in many ways not what the average reader of our time would think of as "poetry." There is nothing romantic (in either sense) about it, no reaching for the sublime or transcendent, no metaphysics, no existential angst, no who-am-I-and-what's-it-all-about-anyway--no introspection, really, of the kind and degree that we begin to find in the poetry that came less than fifty years later.

The passions are certainly there, but are straightforward and down-to-earth--exasperation, indignation--directed at straightforward and down-to-earth causes: the passions of a satirist, in short, which Pope was above all else. What we have in him, as in other 18th century writers, is what was known as "wit," which was something deeper than what we now usually mean by the word: not a gift for clever humor, as in "witty," but  rather a combination of sharply intelligent observation with great skill of expression, as Pope himself explained it in "An Essay On Criticism":

True wit is nature to advantage dress'd,
What oft was thought, but ne'er so well express'd,

And "nature" was not, as we usually mean, the natural non-man-made world--"the environment," but the whole natural order of things, including human nature.

Denunciation of the bad is at least an implicit acknowledgement of the good. But only toward the end of the poem is there a turn toward an explicit defense of it, where Pope defends himself as a friend of virtue, honors his parents, praises Arbuthnot (finally!) and wishes him well--and not too soon, for Arbuthnot died within weeks of the Epistle's publication. 

After all that invective, let's have a touch of the poet's generosity, closing with a brilliant and touching expression of the position, the love, and the duty of child toward elderly parent:

O Friend! may each Domestick Bliss be thine!
Be no unpleasing Melancholy mine:
Me, let the tender Office long engage
To rock the Cradle of reposing Age...

With, I think, a hint of the Resurrection.