Can you pay weekly? I arrive mid-month

Yes, and this is one of the genuinely flexible things about goshiwon. Most will do a partial period.

What is normally available:

1. Pro-rated part month. You arrive on the 18th, you pay for 18th to end of month at a daily rate, then move to normal monthly billing. This is the most common arrangement and most operators offer it without being asked.
2. Weekly. Many buildings do a weekly rate, usually at a slight premium — a week might be a third of a month's rent rather than a quarter.
3. Daily, at some buildings, priced like a cheap guesthouse. Less common but it exists.

How to ask: "18일부터 들어갈 수 있어요? 일할 계산 되나요?" — can I move in from the 18th, is it pro-rated.

For your specific gap, one thing worth considering: those two weeks before your program starts are the best time to view other rooms. Rather than locking into a ten-month arrangement on day one, some students deliberately take a short stay first, use the two weeks to see the neighbourhood and three or four buildings, then commit properly from the 1st. The pro-rated cost of that flexibility is small.

One practical note: arriving on the 18th means your ARC address, bank account and phone all get set up in that window, which is exactly the right use of it. Trying to do all of that in the first week of classes is much worse.

And if the goshiwon you want will not do a partial period, a guesthouse for those two weeks is around 400,000 to 600,000 won and buys you the same flexibility.