Learn English – “put up” meaning

It is slang, and somewhat relies on one knowing that you can preserve things in ethanol, which is the pure form of alcohol (what actually happens is that alcohol is broken down by the liver and converted to acetaldehyde, which is a close relative of formaldehyde (which I'm sure you know is used in embalming). but that's not so much a useful part of the answer as some interesting biochemistry waffle).

In this context, "pickle" is being used in its definition of

preserve (food or other perishable items) in vinegar or brine

and a link is being made between the preservation of food with vinegar, and a human consuming so much alcohol that the resulting acetaldehyde formation would preserve them the same way.

I should point out that the way your sample phrase is worded would imply exaggeration of the amount consumed; it's not literally saying that they've consumed a sufficient quantity to cause that. (as WS2 pointed out, "being pickled" is used as hyperbole for "intoxicated")

This is a separate meaning of the phrasal verb take down, one which is now fairly old-fashioned.

This sense doesn’t seem to be included in many standard dictionaries: Merriam-Webster, American Heritage, and Oxford Dictionaries Online all show no trace of it, and as you mention, nor does The Free Dictionary.

The OED article on take, however, does include it, as sense 8 of the phrasal verb take down:

trans. Of a man: to escort (a female guest) into dinner. Now arch.

1887   Mrs. J. H. Perks, From Heather Hills, II. xviii. 308
      A quiet dinner-party, with a nice, sensible man to take you down.

There are a few other quotes as well, but this one best illustrates the meaning, I think. Quite noteworthy, though: although the article labels this usage archaic and most of the quotes are from the 1800s, the most recent quote is from Jane Feather’s 2011 (!) novel A Wedding Wager, so it is still occasionally used.

While I have no evidence to back this up, my intuitive understanding of the verb is that down here is in essence meant literally: in traditional houses, especially the upper-class settings in the UK where such dinner parties would frequently occur in the 19th century, the dining room or dining hall was located downstairs, while the rooms that people stayed in were upstairs. If you read novels of the time, you will almost invariably see people coming down for breakfast or dinner, and this did literally include walking down the stairs at the time. If a gentlemen were to escort a lady to dinner, they would likely meet on the landing or in the hallways upstairs, and then go down the stairs together.