To the most copious Carminist
of our time, and famous persecutor of Priscian, his
very friend Master Apis lapis : Tho. Nashe wish-
eth new strings to his old tawny Purse, and
all honourable increase of acquain-
tance in the Cellar

It was not unusual for Nashe to subvert the normal style of dedications by adopting the phrases and rhythms of respectful prose, and misapplying them to a patron who was clearly a "good fellow" rather than a pillar of society. He does the same thing in his first-ever dedication, to the clown Will Kempe, which prefaces his first pseudonymous anti-Martinist tract An Almond for a Parrat (1589). Like other Elizabethan writers Nashe resented the lack of patronage he suffered. Ironic dedications to men like William Beeston were one way of alluding to the failure of wealthier men to support the arts. For some speculation on the identity of the dedicatee, select here. For original spelling, select here.

"Veterem etc." :By putting up with an old injury you're asking for a new one.
Nipitaty :A strong beer.
The two phrases of course are not synonymous. The first signifies Nashe's intention to answer Harvey's attack; the second sounds like a drinker's catchphrase. Beeston was evidently not a man of sober habits.
  Gentle M. William, that learned writer Rhenish Wine-and-Sugar, in the first book of his 'Comment Upon Red Noses', hath this saying: "Veterem ferendo iniuriam invitas novam": which is as much in English as "One cup of Nipitaty pulls on another". In moist consideration whereof, as also in zealous regard of that high countenance you show unto scholars, I am bold, instead of new wine, to carouse to you a cup of news - which, if your worship according to your wonted Chaucerism shall accept in good part, I'll be your daily orator to pray that that pure sanguine complexion of yours may never be famished with pot-luck, that you may taste till your last gasp, and live to see the confusion of both your special enemies, small beer and grammar rules.
Sumners: officials of the Archdeacon's court, who summoned people for offences such as drunkeness, swearing and immorality. Nashe is hinting Beeston has paid fines to them - a hint amplified below.  It is not unknown to report, what a famous pottle-pot patron your have been to old poets in your days, and how many pounds you have spent (and as it were, thrown into the fire) upon the dirt of wisdom, called Alchemy : yea, you have been such an infinite Maecenas to learned men, that not any that belong to them (as Sumners, and who not) but have tasted of the cool streams of your liberality.
"...chronicled in the Archdeacon's court" : some time ago Beeston begot children (now grown) on his maids, and was fined for immorality.
"An honest man...etc": see "Harvey".
  I would speak in commendation of your hospitality likewise, but that it is chronicled in the Archdeacon's Court, and the fruits it brought forth, as I guess, are of an age to speak to themselves. Why should virtue be smothered by blind circumstance? "An honest man of Saffron-Walden kept three sons at the University together a long time" ; and you kept three maids together in your house a long time. A charitable deed, and one worthy to be registered in red letters.

Shall I presume to dilate of the the gravity of your round cap, and your dudgeon dagger? It is thought they will make you be called upon shortly to be Alderman of the Steelyard. And that's well remembered: I heard say, when this last Term was removed to Hartford, you fell into a great study and care by yourself to what place the Steelyard should be removed ; I promise you truly it was a deep meditation, and such as might well have beseemed Elderton's parliament of noses to have sit upon.

A tavern in London, only upon the motion, mourned all in black, and forbear to girt her temples with ivy, because the grandam of good fellowship was like to depart from amongst them. And I wonder very much, that you samsoned not yourself into a consumption with the profound cogitation of it.

Diu viuas in amore iocisque, whatsoever you do, beware of keeping diet. Sloth is a sin, and one sin (as one poison) must be expelled with another. What can he do better that hath nothing to do, than fall a-drinking to keep him from idleness?

Fah, me thinks my jests begin already to smell of the cask, with talking so much of this liquid provender.

In earnest thus ; There is a Doctor and his Fart that have kept a foul stinking stir in Paul's Churchyard ; I cry him mercy, I slandered him, he is scarce a doctor till he hath done his acts: this dodipoll, this didapper, this professed poetical braggart, hath railed upon me, without wit or art, in certain four pennyworth of letters and three farthing-worth of sonnets ; now do I mean to present him and Shakerley to the Queen's fool-taker for coach-horses : for two that draw more equally in one oratorical yoke of vainglory, there is not under heaven.

What say you, Master Apis-lapis, will you with your eloquence and credit shield me from carpers? Have you any odd shreds of Latin to make this lettermonger a cockscomb of?

It stands you in hand to arm your self against him; for he speaks against coneycatchers, and you are a coneycatcher, as coneycatching is divided into three parts ; the Verser, the Setter, and the Barnacle.

A Setter I am sure you are not, for you are no musician: nor a Barnacle, for you never were of the order of the Barnardines: but the Verser I cannot acquit you of, for M.Vaux of Lambeth brings in sore evidence of a breakfast you won of him one morning at an unlawful game called rhyming. What lies not in you to amend, play the Doctor and defend.

A fellow that I am to talk with by and by, being told that his father was a rope-maker, excused the matter after this sort; And hath never saint had reprobate to his father? They are his own words, he cannot go from them. You see here he makes a reprobate and a ropemaker, voces convertibiles. Go to, take example by him to wash out dirt with ink, and run up to the knees in the channel, if you be once wetshod. You are amongst grave Doctors, and men of judgment in both laws euery day: I pray ask them the question in my absence, whether such a man as I have described this Epistler to be, one that hath a good handsome pickerdevant, and a pretty leg to study the Civil Law with, that hath made many proper rhymes of the old cut in his days, and deserved infinitely of the state by extolling himself and his two brothers in every book he writes: whether (I say) such a famous pillar of the press, now in the fourteenth or fifteenth year of the reign of his rhetoric, giving money to have his illiterate pamphlet of letters printed (whereas others have money given them to suffer themselves to come in print) it is not to be counted as flat simony, and be liable to one and the same penalty?

I tell you, I mean to trounce him after twenty in the hundred, and have a bout with him with two staves and a pike for this gear.

If he get any thing by the bargain, let whatsoever I write hence-forward be condemned to wrap bumbast in.

Carouse to me good luck, for I am resolutely bent; the best blood of the brothers shall pledge me in vinegar. O would thou hadst a quaffing bowl, which, like Gawain's skull, should contain a peck, that thou mightest swap off a hearty draught to the success of this voyage.

By whatsoever thy visage holdeth most precious I beseech thee, by John Davies' soul and the Blue Boar in the Spittle I conjure thee, to draw out thy purse and give me nothing for the dedication of my Pamphlet.

Thou are a good fellow I know, and hadst rather spend jests than money. Let it be the task of thy best terms to safe-conduct this book through the enemy's country.

Proceed to cherish thy surpassing carminical art of memory with full cups (as thou dost): let Chaucer be new scoured against the day of battle, and Terence come but in now and then with the snuff of a sentence, and Dictum puta, We'll strike it as dead as a door nail ; Haud teruntii estimo, We have cat's meat and dog's meat enough for these mongrels.

However I write merrily, I love and admire thy pleasant witty humour, which no care or cross can make unconversable. Still be constant to thy content, love poetry, hate pedantism. Vade, vale, cave ne titubes, mandataque frangas.