It all depends on what deck and how much you are playing of course, I'd say that you can easily get to rank 5 in first 2-3 days playing tier 1 deck given there were streamers that have done 25 - Legend in 10 days on fresh accounts.
Also I've seen some people getting legends on first days after resets, so as I said in the first sentence ;)
Good luck btw :)
I'm usually finishing around Rank 2-6 every season I find by the end of the first week I'm Rank 10 at least, the next 3 weeks I slowly climb up usually spend a week getting from about 5 to 3 or 2 but I only play casually so I don't dedicate alot of time to playing :)
I would aim to be at rank 1-5 maximum by the 15th, so that gives you half a month to take your time and grind those ranks. The release of an Expansion is an interesting time, you might find a great new deck people don't know how to rect to and climb really fast.
I ususally reach Legend (the 2 times I did, after that I lost interest) in the last 7 days of the month.
It really depends on your time commitment. A lot of people hit legend in the first week, but it gets easier later in the month, as the top players reach legend and filter out of the lower ranks.Rank 5 is basically the halfway point, so if you want to reach legend, a good goal would be to make it to 5 by the end of week 2.
I don't think you should see things that precisely. Sometimes I get stuck at rank 12 for a week and then I reach rank 5 in a day. The real deal is going from rank 5 to legend, and there you have a point: the faster you get to rank 5 the better.
Then you have to follow some basic rules. Obviously knowing the meta helps playing and countering, but you also have to stick to a deck you know a lot of people reached legend with (I'm assuming you're net decking, which is totally fine with me even if I try not to). The point of a deck's winrate is that it's a win/lose ratio, so do not switch decks even if you lose three times in a row (it's bound to happen).
Then you will notice some strange things. With particular decks I get particularly successful depending on the day of the week, or the time of day. It's small things, maybe just coincidences, but that's what got me to Legend the first time, and the few times after that.
Hmm what's the point of anticipating rank at the end of each week tho? It will not affect anything literally, you just play until you get Legend. Also there aren't such "rules" which rank should you be at the end of each week and Ranks have too few stars for anyone to anticipate such thing as it can vary +-3 or even more ranks in a single day, let alone week. The only thing that should concern you is to play until you reach Legend.
There aren't any rules also, it depends on a lot of factors including how good you play, what deck you play and your strategy (if you swap decks like crazy or stick to a good one until you make it) etc. For example, I once needed whole 20 days for Legend from Rank 16 (played 2-3h a day and more on weekends) and the other time I reached it in less than 12 hours total (from Rank 16 to Legend in exactly 11.8 hours). So it really has no rules at all nor that should concern you, just play until you make it