This time, I'd like to make a special shout out to the fine folks at https://scryfall.com/ , which made putting this together FAR easier than it has been in the past.
In terms of the numbers themselves, it's a pretty normal "big" set. 101 commons, 80 uncommons, 53 rares, and 15 mythics. This leads us to .099, .0375, .0165, and .0083 of any particular card of that rarity, respectively, per pack. This gets multiplied by 24 packs to get a per-draft average. If you want to know a about a sealed, you'd divide that by 4. (I'll note that due to the way print runs happen, I think there's one common with a slightly different incidence rate, but there's little way at this point to know which that is; I'm also ignoring foils here, since I'm not sure how that replacement works, so that would slightly increase non-commons and decrease commons; these are all very small differences, but I wanted to mention them in the interest of full disclosure).
FixingOne of the first things I always want to look at in any format is how much mana-fixing there is. This helps us figure out how many colors we can be playing, how much you'd have to work for extra colors, how easy it is to splash, how much contempt you should have for picking multicolor cards early, etc.
Amonkhet has 4 common mana fixers, 1 uncommon, and 9 rares (I'm not including Vizier of the Menagerie, which only fixes for creatures). This leads us to a total of an average of 14 pieces of fixing per draft. Typically you want something like 4-8 pieces of fixing to play a third color, which means you'd need roughly half (or maybe a little under) of the fixing in the draft - seems possible, but you'd have to work for it. But let's drill a bit deeper. Painted Bluffs is a common fixer that could go in any deck, but not one you'd want to. Cascading Cataracts and Pyramid of the Pantheon are similar, but at rare. The cycling lands are probably going to be quite hard to pick up if you don't open them, and in any case will only fix your mana if you just happen to be the right colors. This leaves us with Evolving Wilds as the only good, reliable fixer for any colors, which is a place we've been pretty often before. Additionally in this set, though, we're back to having noticeably more fixing in Green exactly - Oashra Cultivator and Gift of Paradise at common, Spring of Spring//Mind at Uncommon, and a couple different rares all add up to make Green the color of fixing again. It's worth noting that these are generally a bit overpriced from what we'd expect (3 mana Rampant Growth seems to be the norm here), but will get the job done in a pinch. And importantly, splashing multiple colors seems only marginally harder than splashing one, and actually easier than trying to be fully 3 colors.
CyclingSure, cycling is a theme of the set. But just how present is it? EVERYWHERE. There are fully 20 Commons, 10 Uncommons, and 8 Rares with the popular returning mechanic, leading to an average of 59.7 cards per draft! This means even the average player will end up with 7-8 of these cards in their pool. And some of those won't be in the right colors, and some will be unplayable (though the option to cycle means very few will be embarrassingly bad). But even if your normal half-the-cards you draft end up in your deck, you're still looking at about 4 per player. Which means if you crack open a Drake Haven, and you actually prioritize these cards a bit, you should really be able to have plenty of enablers to turn that card on. I'll also note here that most of these cards that care about cycling also trigger off of other forms of discard, of which there are 14 in the set - bringing you to an even healthier number of enablers. So you shouldn't really have problems in 'getting there' with those kinds of cards.
How many such rewards are there? Well, if you also include cards like Shadow of the Grave and Sacred Excavation, which don't trigger off cycling per se, but definitely care about the mechanic, you end up with 3 commons, 6 uncommons, and 5 rares, for a total of 14.5 per draft. So not all that many. When you factor in that a lot of these are at higher rarity, and several of the commons give mediocre bonuses, I don't think this is an archetype you should expect to see in every draft pod. But it is something you can go with if you get the right card(s) early. And worth noting that this is centred in blue and black particularly, also with some presence in red.
Lastly, because cycling is something that happens from the hand, at instant speed, and is on lots of cards, if your opponent has something like Hekma Sentinels or Pitiless Vizier, keep in mind that they basically have threat-of-activation on activated abilities - since most any card in hand could be a combat trick with card advantage. So value that accordingly in the draft, and play round or bluff it accordingly in gameplay.
Embalm appears on 5 commons, 4 uncommons, 5 rares, and a mythic. It is centred mostly in white, with strong representation in blue as well, and the smallest sprinkles in Red and Green. In total, you can expect 17.7 Embalm creatures to show up on average in a draft. Because of the color imbalance, you can expect white and blue drafters to probably have a few each (WU drafters a bit more than that even), but not at all a strongly themed deck.
ZombiesThis leads us right to Zombies, which seem to be the tribe du jour on Amonkhet. Apart from the Embalm cards (all of which make white zombie tokens when embalmed), there are 28.75 other zombies per draft in the set, (including cards which make multiple zombie tokens, like Liliana or her Mastery, once each for their rarity). Altogether, that makes a total of 46.4 - definitely less than cycling, but more than about anything else you're going to find. Especially important is that these other zombies are all white and/or black, so that when you combine the embalm in, you get the most Zombies in white, followed by black and blue, and very few in red or green.
But the bigger story here is the pay-offs for zombies. There appear to be quite a few in the set. But the problem is that, like with the cycling bonuses, they're focused at higher rarities. 2 commons, 4 uncommons, and 2 rares leaves you with only 9.1 zombie bonus cards per draft (I didn't count the Liliana ultimate here, full disclosure). So this is somewhat like the BW Lifegain theme from Oath of the Gatewatch - sometimes it will come up, but you can have decks even in those colors where it doesn't really.
"Heckbent"Something that people have been noticing throughout the spoilers is that there seems to be a subtheme of cards, mostly in black and red, which care about having few cards in hand - specifically, many of them are improved when you get to having 0-1 cards in hand. People have dubbed this "Heckbent" as a lite version of the Hellbent (no cards in hand) keyword from Dissension. But this is really a constructed-slanted mechanic - 1 common, 2 rares, and a mythic have that text, plus an extra uncommon that's huge but shrunken for each card in hand. Don't count on this in limited.
-1/-1 CountersInstead of the near-ubiquitous run of +1/+1 counter mechanics we've had over the last few years, this block returns us for the first time since Scars block to -1/-1 counters. These are fairly prevalent in the set, with 26.7 cards per draft that give them out. These are primarily in black and green, with a bit in red. And it's especially worth noting that many of these cards actually have you putting the counters on your own creatures, at least at first (many of those in turn have ways for you to take them off later).
How many cards care about these kinds of counters is, as often, the bigger question. The answer in this case is 12.4 per draft (this follows some logical progression on what counts as "caring about" - I'm not counting here Exemplar of Strength, but of course I am counting Nest of Scarabs). This is definitely the kind of thing which again, doesn't look terribly supported, but again, is something you probably will see from time to time.